You have engineered the X chromosomes in female mice such that one X chromosome expresses green fluorescent protein when active, while the other expresses red fluorescent protein. You have used these mice to study cancer in females. You know that each tumor is a clonal cellular proliferation, meaning all of its proliferating cells are descendants of a single original cancer-causing cell. It follows that, unless X-chromosome inactivation is perturbed in tumors, ...
a) all tumor cells in one mouse should express the same fluorescent protein (either red or green), but tumor cells from different mice can show either red or green fluorescence.
b) the cells in any tumor should all express the same fluorescent protein (either red or green), but independently derived tumors in the same mouse can show either green or red fluorescence.
c) different cells within each tumor can express different fluorescent proteins, and the tumors would therefore show yellow fluorescence, but each cell shows either red or green fluorescence.
d) each cell can express both fluorescent proteins and would therefore emit yellow fluorescence, and the tumors would glow in yellow as well.
e) different tumors would show red, yellow, or green fluorescence.

If the supply of clean water was reduced as the global population increased, it would have the most significant impact on humans as a biotic factor. Clean water is essential for human survival and is required for various activities, including drinking, sanitation, and food production.

As the population increases, the demand for water also increases. However, the availability of clean water resources may not keep pace with the growing population. This can lead to water scarcity, which can have severe consequences for human health and well-being.

Reduced water supply can result in inadequate access to safe drinking water, leading to waterborne diseases such as cholera, typhoid, and dysentery. Insufficient water for sanitation purposes can also contribute to poor hygiene and the spread of diseases.

In regions where water scarcity is already a problem, population growth can exacerbate the situation, putting increased pressure on limited water resources. Therefore, the reduction in the supply of clean water would have a significant impact on human populations as the global population increases.

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The correct order of organs that act when someone decides to pick up a cup after seeing that it is about to spill is: A. eye, brain, muscle. The eye first detects the visual cue that the cup is about to spill and sends this information to the brain.

The answer is A. eye, brain, muscle.

The brain then processes this information and decides that action is needed to prevent the spill. Finally, the brain sends signals to the appropriate muscles, which then contract and move the hand to pick up the cup. This process involves a complex interplay of sensory and motor neurons in the body, and it can vary slightly depending on the individual and the specific situation.

The correct order of the organs involved in this process is: A. eye, brain, muscle.

1. The eye sees the cup about to spill.
2. The brain processes the information and decides to pick up the cup.
3. The muscles are signaled by the brain to perform the action of picking up the cup.

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The structure and maintenance of all ecosystems result from the totality of organisms in that ecosystem. Ecosystems are made up of different organisms that interact with each other in various ways.

Each organism has a unique role in the ecosystem, and they are all interconnected. The interactions between organisms and their environment shape the structure and maintenance of the ecosystem. For example, predators and prey interact to maintain a balance in the ecosystem, while decomposers break down dead organisms and recycle nutrients.
While the types of plants and animals in an ecosystem play a crucial role, they are not the only factors that determine its structure and maintenance. Bacteria, fungi, and soils also play important roles in nutrient cycling, decomposition, and maintaining the health of the ecosystem. Therefore, the answer is not A or D, as the types of plants or animals do not solely determine the ecosystem's structure and maintenance.
Overall, the totality of organisms in an ecosystem, including plants, animals, bacteria, fungi, and soils, all contribute to its structure and maintenance. The interactions between these organisms and their environment shape the ecosystem and ensure that it continues to thrive.

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Bone serves as a mineral reserve for calcium and phosphate ions.

Calcium is an essential mineral for various physiological processes in the body, including bone formation, nerve function, muscle contraction, and blood clotting. It plays a vital role in maintaining the integrity and strength of bones. When calcium levels in the blood are low, the body can release calcium from the bone matrix to maintain the necessary calcium levels for these physiological functions.

Phosphate, in the form of phosphate ions (PO43-), is also stored in bone. Phosphate is an important component of adenosine triphosphate (ATP), which is the primary energy source in cells. It is also a crucial component of DNA, RNA, and several important molecules involved in cellular signaling and metabolism.

Both calcium and phosphate ions are deposited and stored in the mineralized matrix of bone, primarily in the form of hydroxyapatite crystals. These mineral reserves can be mobilized as needed by the body to maintain appropriate levels of calcium and phosphate in the bloodstream and support various physiological processes.

In summary, bone serves as a mineral reserve for calcium and phosphate ions, which are essential for numerous biological functions in the body.

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"RDA" typically stands for Registered Dental Assistant, and the fulcrum technique is commonly used in dental settings for stability and control during various procedures, such as scaling and root planing. However, it is not specifically associated with coronal polishing.

Coronal polishing is a dental procedure performed to remove stains and plaque from the surfaces of teeth. It is typically carried out by a dental hygienist or a dentist. During coronal polishing, a slow-speed handpiece with a prophy cup or a brush is used to apply a polishing agent to the teeth. The polishing agent, combined with the rotational movement of the handpiece, helps remove surface stains and plaque.

While the fulcrum technique is essential for maintaining stability and control during various dental procedures, its specific relevance to coronal polishing may vary depending on the individual and their technique.

It's best to consult with a dental professional or reference dental guidelines for specific recommendations on performing coronal polishing and the use of the fulcrum technique.

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The baroreceptors sense a decrease in blood pressure which leads to an increase in sympathetic output and a decrease in parasympathetic output. This results in an increase in SA node firing and vasoconstriction of veins. Therefore, the correct option is ↑ sympathetic output & ↓ parasympathetic output →↑ SA node firing & vasoconstrictor veins.

The cardiovascular control system (CVCC) is responsible for regulating blood pressure and maintaining adequate blood flow to tissues. Baroreceptors are specialized cells located in the walls of blood vessels that detect changes in blood pressure. When blood pressure decreases, the baroreceptors signal the CVCC to increase sympathetic output and decrease parasympathetic output. This results in an increase in heart rate (due to increased firing of the sinoatrial or SA node) and vasoconstriction of veins (which increases venous return to the heart). These responses help to increase blood pressure and restore blood flow to the injured tissue.

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The fungi that cause serious plant diseases such as Dutch elm disease and chestnut blight belong to the phylum Ascomycota. This phylum includes over 64,000 species, including many important plant pathogens and symbionts.

Ascomycetes are characterized by the formation of ascospores within a specialized sac-like structure called an ascus, which is a defining characteristic of the phylum.
The fungi that cause Dutch elm disease and chestnut blight are both examples of ascomycetes. Dutch elm disease is caused by the fungus Ophiostoma ulmi, while chestnut blight is caused by the fungus Cryphonectria parasitica. These fungi are both devastating plant pathogens that have caused significant economic and ecological damage.
Overall, Ascomycota are a diverse and important group of fungi that play key roles in both plant and animal ecosystems. While many species are important plant pathogens, others are important decomposers, symbionts, and producers of valuable products such as antibiotics and food additives.

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Diphyllobothrium latum is not a helminth that causes primarily intestinal symptoms. It is a type of tapeworm that is acquired by eating undercooked or raw fish infected with its larvae.

The larvae can then develop into adult tapeworms in the small intestine. While it can cause gastrointestinal symptoms such as abdominal pain, diarrhea, and nausea, it is primarily associated with vitamin B12 deficiency anemia due to its ability to absorb the vitamin from the host's intestine. Trichuris trichiura, Enterobius vermicularis, and Taenia solium are all helminths that primarily cause intestinal symptoms. Trichuris trichiura, also known as the whipworm, can cause abdominal pain, diarrhea, and bloody stools. Enterobius vermicularis, or the pinworm, can cause itching around the anus and sometimes abdominal pain. Taenia solium, or the pork tapeworm, can cause abdominal pain, diarrhea, and weight loss. It can also cause more serious symptoms if the larvae migrate to other organs such as the brain.

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Male castration would result from the operation known as bilateral orchiectomy. Bilateral orchiectomy is the surgical removal of both testicles.

Bilateral orchiectomy, resulting in the elimination of testosterone production and rendering the individual sterile. This procedure is commonly performed as a treatment for testicular cancer or as a form of hormone therapy for conditions such as prostate cancer or gender reassignment. The removal of both testicles leads to the cessation of sperm production and the loss of male reproductive function.

It is important to note that unilateral orchidectomy refers to the removal of only one testicle, which does not result in castration since the other testicle can still produce testosterone. Vasectomy, on the other hand, is a different procedure that involves the cutting or blocking of the vas deferens, a duct that carries sperm, but it does not result in castration. Bilateral oophorectomy, meanwhile, refers to the removal of both ovaries and is a procedure performed in females, not males.

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The residence time of carbon in the soil organic carbon pool refers to the average duration that carbon remains stored in the soil before it is released back into the atmosphere or transferred to another reservoir.

The residence time of carbon in the soil organic carbon pool can vary depending on various factors such as climate, vegetation type, soil composition, and management practices.

On average, carbon can reside in the soil organic carbon pool for several decades to centuries. However, it's important to note that carbon turnover rates can vary between different soil types and ecosystems.

During its residence time in the soil organic carbon pool, carbon can be actively involved in various processes, including decomposition of organic matter, microbial activity, plant uptake, and carbon sequestration.

Changes in environmental conditions and human activities can influence the residence time by affecting the balance between carbon inputs and outputs in the soil.

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In precipitation tests, maximum precipitation occurs when the amount of the antibody and the amount of the antigen are at optimal proportions.

The precipitation test is a method used to detect the presence of specific antigens in a sample by measuring the degree of antigen-antibody interaction. In the test, a solution containing the antigen is added to a solution containing the antibody. If the two bind together, they will form a complex that can precipitate out of solution. The amount of precipitation formed is proportional to the amount of antigen present. However, if the amount of antibody exceeds the amount of antigen, the precipitation reaction will not be as strong as when the two are at optimal proportions. On the other hand, if a complex solution of many antibodies is used, the precipitation reaction may not be as specific as when a single type of antibody is used. Therefore, the best conditions for maximum precipitation in a precipitation test are achieved when the amount of the antibody and the amount of the antigen are at optimal proportions.

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The person must leave the money in the bank for approximately 19.8 years until it reaches $15700.

By using the compound interest formula and substituting the given values, we calculated the time it would take for the investment to grow from $6500 to $15700 at an interest rate of 6.75% compounded semi-annually. The result was approximately 19.8 years. This means that if the person keeps the money in the bank for this duration, the investment will accumulate enough interest to reach $15700.

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Eukarya cannot initiate transcription in the middle of an mRNA due to a. they rely on the 5' cap for initiation.

The 5' cap, a modified guanine nucleotide, is crucial for ribosome recognition and binding. Eukaryotic ribosomes are unable to bind to the middle of eukaryal mRNAs without this cap. Additionally, eukaryal mRNAs typically only contain one Shine-Dalgarno sequence, which is essential for ribosome binding in prokaryotes. This sequence is not commonly found in eukaryotes, further limiting ribosome binding possibilities in the middle of an mRNA.

Finally, eukaryal ribosomes move at a rapid pace during translation, which hampers the ability of new ribosomes to bind in the middle of an mRNA. This fast movement ensures efficient translation but restricts the potential for transcription initiation at internal sites. In summary, eukarya cannot initiate transcription in the middle of an mRNA due to their reliance on the 5' cap for initiation, the presence of only one Shine-Dalgarno sequence, and the rapid movement of eukaryal ribosomes during translation.

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The psychoactive plant that is a dioecious annual with distinctive palmately compound leaves is Cannabis sativa. Cannabis sativa is one of the three main subspecies of the Cannabis plant, the other two being Cannabis indica and Cannabis ruderalis.

Cannabis sativa is known for its tall, thin structure and its ability to grow up to 20 feet in height. It is primarily grown for its psychoactive properties, which are due to the presence of delta-9-tetrahydrocannabinol (THC) in its flowers and leaves.
Cannabis sativa has been used for medicinal and recreational purposes for thousands of years. It has been shown to have a variety of therapeutic effects, including pain relief, reducing anxiety and depression, and improving sleep. However, it is also associated with some negative effects, such as impaired coordination and memory, and may have adverse effects on mental health in some individuals.
Overall, Cannabis sativa is a widely cultivated and consumed plant, with both positive and negative effects. Its psychoactive properties have led to its use for recreational and medicinal purposes, but it is also subject to legal and social controversy in many parts of the world.

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Derivatives of the mesoderm include vertebrae and ribs, glandular derivatives of the digestive tract, and epithelium of the digestive tract. Option d is correct answer.

The mesoderm is one of the three primary germ layers formed during embryonic development. It gives rise to various structures and tissues in the body. While it does not include all nervous tissue (which is derived from the ectoderm), derivatives of the mesoderm do include vertebrae and ribs, glandular derivatives of the digestive tract, and epithelium of the digestive tract.

Vertebrae and ribs are formed from mesodermal cells during embryonic development. They contribute to the skeletal system and provide support and protection for vital organs. Glandular derivatives of the digestive tract, such as the liver and pancreas, Submucosa also arise from the mesoderm. These glands play important roles in digestion and metabolism. Additionally, the mesoderm gives rise to the epithelium of the digestive tract, which lines the inner surface of the gastrointestinal system and is involved in nutrient absorption and secretion.

Therefore, derivatives of the mesoderm include vertebrae and ribs, glandular derivatives of the digestive tract, and epithelium of the digestive tract.

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The amino acid sequence produced during translation is given as:

Sequence 1: 5'-AUGGACGAAUAGCCUGA-3 - met-thr-asn-tyr-arg-(stop)

Sequence 2: 5'-AUGGACGAAUAUGCCUGA-3' - met-thr-asn-(stop)

The first column of the chart represents a codon's initial base; remain in this row. Stay in the box marked by the second base in the centre of the chart. The amino acid that matches the mRNA codon can be found by looking at the third base in the far right column.

Now that we have read our mRNA transcript and used the provided table, we know the precise order of the amino acids. Methionine, valine, histidine, and leucine are the codons that will be translated into when reading from the five prime to three prime direction.

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Terran can manipulate several factors to differentially regulate the multiple operons involved in the starvation response in E. coli. One such factor is the presence or absence of specific nutrients.

By providing or withholding certain nutrients, Terran can activate or deactivate specific operons that are involved in the response to starvation. Another factor that Terran can manipulate is the temperature at which the E. coli strain is grown. Different temperatures can affect the expression of certain genes and operons, so by adjusting the temperature, Terran can selectively regulate the response to starvation.
Terran can also manipulate the concentration of signaling molecules such as cyclic AMP (cAMP) and guanosine tetraphosphate (ppGpp) that are involved in regulating gene expression in E. coli. These molecules can activate or repress specific operons in response to environmental changes such as starvation. Terran can also use genetic engineering techniques such as creating mutations or deletions in specific genes or operons to alter their expression in response to starvation conditions.
Overall, Terran has several options to selectively regulate the multiple operons involved in the starvation response in E. coli. By manipulating factors such as nutrients, temperature, signaling x, and gene expression, he can enhance the survival of the E. coli strain in starvation conditions.

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The verbal description of the storyboard contains two problems: the incorrect use of the term "adapted" and the assumption that freckled Bobbits dominated over non-freckled Bobbits solely because of blending in with the speckled sand. It is important to address these issues to ensure accurate understanding of the concept of natural selection and the factors contributing to the dominance of certain traits.

The problems can be fixed by clarifying the concept of adaptation and considering additional factors that may have influenced the dominance of freckled Bobbits.

Problem: The incorrect use of the term "adapted." In the verbal description, it states that "as each Bobbit adapted, freckled Bobbits dominated over the non-freckled Bobbits." This usage of the term implies that individual Bobbits actively adapt in response to their environment, which is not how adaptation works. Adaptation occurs over generations through the process of natural selection, where individuals with advantageous traits have higher survival and reproductive success.

Solution: To fix this problem, the description should be modified to reflect that freckled Bobbits were more likely to survive and reproduce due to their advantageous trait of blending in with the speckled sand, leading to a higher proportion of freckled Bobbits in subsequent generations.

Problem: Assuming that freckled Bobbits dominated solely because of blending in with the speckled sand. While blending in with the environment can provide a selective advantage, it is important to consider other factors that could have influenced the dominance of freckled Bobbits, such as predation, mate preference, or other environmental pressures.

Solution: To address this problem, the verbal description should be expanded to acknowledge that while blending in with the speckled sand may have contributed to the fitness of freckled Bobbits, additional factors may have also played a role. This includes factors like reduced predation risk or increased chances of successful reproduction through mate selection.

By fixing these problems, the verbal description will provide a more accurate representation of natural selection, emphasizing the gradual changes in populations over time and the interplay between advantageous traits and environmental factors.

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Separating a complex culture of bacteria into distinct colonies on solid media is achieved using a technique called streaking.

Streaking is a method of isolating individual bacterial cells from a mixed culture and growing them into separate colonies. This is done by spreading the cells over a solid agar plate in a series of lines. As the cells grow, they will form individual colonies that can be easily identified and isolated.

To streak a plate, you will need the following materials:

A sterile inoculating loop

A sterile petri dish

A nutrient agar plate

A bacterial culture

Instructions:

Label the petri dish with the name of the bacterial culture.

Flame the inoculating loop to sterilize it.

Dip the loop into the bacterial culture.

Make a small streak across the surface of the agar plate.

Flame the loop again.

Repeat steps 4 and 5, making each streak slightly closer to the center of the plate.

Incubate the plate at 37 degrees Celsius for 24-48 hours.

After incubation, you will see individual colonies of bacteria growing on the plate. Each colony is a clone of the original bacterial cell that was streaked onto the plate. You can then use these colonies to perform further experiments or to identify the bacteria.

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The frequency of a detrimental allele will decrease more rapidly over time if it acts in a dominant fashion compared to if it acts in a recessive fashion.

This is because a dominant allele will always be expressed in the phenotype of individuals that carry it, whereas a recessive allele will only be expressed if it is present in a homozygous state. In the case of a dominant allele, individuals that carry it and express the detrimental phenotype are more likely to be eliminated by natural selection, which reduces the frequency of the allele in the population.

In contrast, individuals that carry a recessive allele but do not express the detrimental phenotype can still pass it on to their offspring, which slows down the rate at which the frequency of the allele decreases in the population.

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Genetic modification is the process of changing an organism's genetic material or gene composition to achieve a specific goal.

Scientists can use several methods to modify the genetic makeup of an organism. The CRISPR-Cas9 gene-editing technique is one of the most powerful methods. Gene modification can be used to create structural changes in the genes of the African elephant. Once the structural change has been made to the genes responsible for tusk growth, it would affect the structure and function of the African elephant. In this case, the pretend change would be to increase the thickness of the tusks. As a result, the elephant's tusks would grow larger and thicker than normal.

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"Plants use photosynthesis to make sugars and deliver oxygen as a result. Using oxygen in the air, animals use cellular respiration to forge energy from sugars they consume from plants or animals that eat plants.

This assertion portrays the corresponding connection between photosynthesis and cell breath. Animals use cellular respiration to generate energy and produce carbon dioxide and water, while plants use photosynthesis to produce glucose and oxygen. The course of photosynthesis and cell breath supports life on Earth by keeping harmony between oxygen and carbon dioxide in the climate.

Oxygen and glucose are produced through photosynthesis, which uses water and carbon dioxide. The plant consumes glucose as a food source, producing oxygen as a byproduct. Oxygen and glucose are transformed into carbon dioxide and water during cellular respiration. Byproducts include carbon dioxide and water, and energy transformed by the process is ATP.

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The organs that control the movement of molecules in the body are primarily the circulatory system, which includes the heart and blood vessels, and the respiratory system, which includes the lungs. These organs work together to transport molecules such as oxygen, nutrients, hormones, and waste products throughout the body.

The circulatory system, composed of the heart, blood vessels, and blood, plays a vital role in the movement of molecules. The heart acts as a pump, continuously circulating blood throughout the body. Blood vessels, including arteries, veins, and capillaries, transport oxygen, nutrients, hormones, and other molecules to different tissues and organs. This system also helps remove waste products such as carbon dioxide and metabolic byproducts.

The respiratory system, consisting of the lungs and associated structures, facilitates the movement of gases, particularly oxygen and carbon dioxide. Oxygen is inhaled into the lungs and diffuses across the lung tissue into the bloodstream, where it binds to red blood cells for transport to body tissues. Carbon dioxide, a waste product, is carried back to the lungs through the bloodstream and exhaled from the body.

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During the acquisition step in the CRISPR/Cas defense against viral infection in bacteria, the bacteria incorporate fragments of viral DNA or RNA into their own genome. This process allows the bacteria to "remember" the viral genetic material and mount a specific defense against future infections by the same or closely related viruses.

1. Viral Infection: Bacteria get infected by a specific virus (also known as a bacteriophage) that carries its genetic material in the form of DNA or RNA.

2. Recognition: The CRISPR/Cas system in the bacteria recognizes the invading viral DNA or RNA as foreign.

3. DNA Fragmentation: The bacterial CRISPR-associated (Cas) proteins cleave the viral DNA or RNA into smaller fragments.

4. Spacer Acquisition: The bacteria select a small fragment of the viral genetic material and incorporate it into their own genome. This small fragment is known as a spacer.

5. Integration: The spacer is integrated into a specific region of the bacterial genome known as the CRISPR array. The CRISPR array consists of multiple spacers separated by repetitive DNA sequences.

6. Memory: The CRISPR array with the newly acquired spacer becomes a heritable memory of the viral infection, allowing the bacteria to recognize and respond to the same or similar viruses in the future.

7. Adaptation: The CRISPR/Cas system, with the help of other proteins, ensures the accurate insertion of the spacer into the CRISPR array and protects it from degradation.

By incorporating spacers derived from viral genetic material, bacteria can use the CRISPR/Cas system to recognize and destroy specific viral DNA or RNA sequences during subsequent infections. The acquired spacers serve as a genetic memory that allows the bacteria to mount a targeted immune response against viral invaders.

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To prevent extinctions, several ideas can be implemented. Firstly, conserving and protecting natural habitats is crucial.

This involves creating and maintaining national parks, wildlife reserves, and protected areas to safeguard diverse ecosystems. Secondly, strict regulations should be enacted to combat illegal wildlife trade, poaching, and habitat destruction. Education and awareness programs can play a pivotal role in promoting sustainable practices and raising public consciousness about the importance of biodiversity conservation. Additionally, supporting scientific research and initiatives aimed at species recovery, such as captive breeding programs and habitat restoration projects, can aid in preserving endangered populations.

Collaboration between governments, organizations, and communities is essential for effective conservation strategies and the implementation of sustainable development practices to mitigate the factors contributing to species extinctions.

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Animals eat plants or other animals that have eaten plants and therefore contain carbon. hence option C is correct.

Carbon enters the ecosystem through photosynthesis in plants, which converts carbon dioxide from the air into organic molecules. Animals then consume these plants (or other animals that have consumed these plants), and in doing so, they obtain the carbon-containing organic molecules that they need to survive.

This process is known as the carbon cycle, and it is the primary way that carbon is transferred from the abiotic (non-living) environment to the biotic (living) environment. Therefore, option C is the most accurate statement regarding how carbon might make its way into animals in the ecosystem.

Hence option C is correct.

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To polarize naïve T cells to the TH1 lineage in vitro, you would choose IL-12 as the reagent. IL-12 is a cytokine that plays a crucial role in promoting the differentiation of naïve T cells into TH1 cells. It stimulates the production of IFN-gamma, a key cytokine associated with the TH1 immune response.

IL-12 acts by binding to its receptor on naïve T cells, initiating signaling pathways that promote TH1 differentiation. By providing exogenous IL-12 during T cell activation, you can enhance the development of TH1 cells.

The other reagents listed have different effects or are associated with different T cell lineages. Anti-TCR antibody and anti-CD28 antibody are used for T cell activation but do not specifically drive TH1 polarization.

CTLA-4 Ig and anti-CD80 antibody modulate co-stimulatory signals but are not directly involved in TH1 differentiation. IL-4 and IL-17 are associated with TH2 and TH17 lineages, respectively, which are distinct from TH1. Therefore, IL-12 would be the appropriate choice for polarizing naïve T cells to the TH1 lineage.

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The individuals that are actually observed within this subset are referred to as the observed sample. This sample is used to make inferences about the larger population from which it is drawn.

In statistical terms, a subset of a population refers to a smaller group of individuals that is selected from a larger population based on specific criteria. This subset is often chosen for research or study purposes to represent the larger population and draw conclusions about it.
It is essential to choose a sample that is representative of the population to obtain accurate results.
The observed sample is a crucial element in statistical analysis as it provides information on the characteristics of the population being studied. By observing the individuals within the subset, researchers can draw conclusions about the population's behavior, characteristics, and attitudes.
In summary, a subset of a population refers to a smaller group of individuals chosen from a larger population, and the individuals that are observed within the subset are called the observed sample. This sample is used to draw conclusions about the population's behavior and characteristics and is crucial in statistical analysis.

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Answer: For the first trait 50% of the offspring will end up with the Dominant trait, you didn’t specify what the traits were. For the second one it’s 100%, if you need help with genotype and phenotype here’s an additional reference plus my work on the problem