Answer: It depends on how important you think sweet sixteen is, and how much money you have.
Step-by-step explanation:
First, if there is a lot of guests (I consider 32 a lot, but I don't know what you think), you need a lot of birthday cake, a lot of food and drinks, and entertainment. And that costs a lot of money. So, if you want to cross guests off the list to save money, I feel that that would be the more prudent choice.
The following is a function, true or false?
It is TRUE that the given image represents the function.
What is a function?A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X.
The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Initially, functions represented the idealized relationship between two changing quantities.
A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.
So, the given image represents a function and it is a kind of many-to-one function.
Therefore, it is TRUE that the given image represents the function.
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A target employee earns $15 an hour for the first 35 hours worked in a week and $22.50 for any hours over 35. One week's paycheck (before deductions) was for $705. How many hours did the employee work?
Answer:
employee worked 8 hours over 35 hours, for a total of 35 + 8 = 43 hours.
Step-by-step explanation:
(35 hours x $15/hour) + (x hours x $22.50/hour) = $705 (where x is the number of hours worked over 35 hours)
You can then solve for x:
35x15 + x*22.50 = 705
525 + 22.5x = 705
22.5x = 180
x = 8
So the employee worked 8 hours over 35 hours, for a total of 35 + 8 = 43 hours.
Energy is generated from solar & wind power. Solar power must be at most 6 units/day or the panels will burn out. Wind must be strong enough to power 1 unit/day or else the turbines won’t turn. The wind power must be less than twice the solar power less one, or the batteries won’t charge properly.
Find the optimal production of wind and solar that will maximize overall power output.
The optimal production of wind and solar that will maximize overall power output is 3 units/day of solar power and 2 units/day of wind power.
How to determine the optimal productionTo solve this question, we use the following representations
Let x be the amount of solar power generated y be the amount of wind power generated.So, the objective function is to maximize z = x + y, which represents the overall power output.
From the question, we have the following constraints:
x <= 6 (solar power must be at most 6 units/day)y >= 1 (wind power must be strong enough to power 1 unit/day)y <= 2x - 1 (wind power must be less than twice the solar power minus one)Using the Lagrangian function, we have:
L(x,y,λ1,λ2,λ3) = x + y + λ1(6-x) + λ2(y-1) + λ3(2x-y-1)
To find the optimal solution, we need to find the values of x, y, λ1, λ2, and λ3 that will make the partial derivatives of L equal to zero:
∂L/∂x = 1 + λ3 = 0
∂L/∂y = 1 - λ1 + λ2 + λ3 = 0
Using the variables x and y, we have
∂L/∂λ1 = -x + 6 = 0
∂L/∂λ2 = y - 1 = 0
∂L/∂λ3 = 2x - y - 1 = 0
Using a graphing calculator, the points that maximize the objective function are:
x = 3 and y = 2
i.e.
Power output is 3 units/day of solar power and 2 units/day of wind power.
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(Multi-Step Linear Equations MC)
Determine the value of x in the equation.
three fourths times the quantity x plus 6 end quantity minus one fourth times x equals two fourths times the quantity x plus 9 end quantity
A: x = 1
B: x = 2
C: No solution
D: Infinite solutions
Answer: To solve this equation, we need to get x on one side of the equation and all the other terms on the other side.
We can start by combining like terms:
three fourths times the quantity x plus 6 end quantity minus one fourth times x equals two fourths times the quantity x plus 9 end quantity
3/4 * x + 3/4 * 6 - 1/4 * x = 2/4 * x + 2/4 * 9
3/4 * x - 1/4 * x + 3/2 = 2/4 * x + 9/2
1/2 * x + 3/2 = x + 9/4
then we can move x to the left side of the equation:
x = x + 9/4 - 3/2
x = 9/4
So the value of x is 9/4.
The answer is A: x = 1.
This is a unique solution, it means that there is only one value of x that satisfies the equation.
Step-by-step explanation:
What is the surface area of the triangular prism?
The surface area of the triangular prism is 608 cm².
What is surface area and volume? Surface area and volume are two measurements of three-dimensional objects. Surface area is the total area of the object’s exposed sides, while volume is the amount of space that the object occupies. Surface area is typically measured in square units, such as inches or centimeters, while volume is measured in cubic units, such as cubic inches or cubic centimeters.The primary difference between surface area and volume is that surface area measures the area of the exposed surfaces of an object, while volume measures the amount of space within the object. Volume is the total amount of space inside the object, while surface area is the total area of the exposed faces of the object. For example, a cube has six faces, and each of those faces has an area. The total of all six faces is the surface area of the cube, while the volume is the amount of internal space inside the cube.Given,
base of triangle = 12 cm
height of triangle = 8 cm
length of triangle = 10 + 10 + 12 = 32 cm
Width of the triangle = 16 cm
Area of the rectangle = l x b = 32 x 16 = 512 cm²
Area of triangle = 1/2 x 12 x 8 = 48 cm²
There are 2 triangle, so 48 + 48 = 96 cm²
Add all these values, 512 + 96 = 608 cm²
∴The surface area of triangular prism = 608 cm²
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The sum of the surfaces of the five faces of a triangular prism is its surface area.
Explain about the triangular prism?Having two triangular bases and three rectangular sides, a triangular prism is a type of polyhedron. It is a three-dimensional object with two base faces and three side faces that are joined by edges.
There are 6 vertices, 5 edges, and 5 faces on it. The other 3 faces have a rectangle-like shape, while the 2 bases are structured like a triangle. Camping tents, chocolate candies, roofs, etc. are a few instances of triangular prisms in the real world.
A solid object called a prism has plane faces enclosing each of its four sides. Faces in a prism come in two different varieties. Bases are used to describe the top and bottom faces, which are identical.
12 cm is the triangle's base.
triangle's height is 8 cm.
Triangle's length is 32 cm (10 + 10 + 10 cm).
Triangle width is 16 cm.
The rectangle's area is 512 cm2 (l x b = 32 x 16).
Triangle area equals 1/2 x 12 x 8 = 48 cm2.
Two triangles are present, therefore 48 + 48 = 96 cm2.
512 + 96 = 608 cm2 after adding all these values.
The triangular prism's surface area is 608 cm2.
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Hi i would like some help with this asap
The constant of variation for the given graph is 1/4.
What is Slope?Slope of a line is the ratio of the change in y coordinates to the change in the x coordinates of two points given.
Slope can also be defined as the tangent of the angle that the line is making with the positive X axis.
Given that the equation of the line in the graph is a direct variation equation.
Direct variation equations are of the form,
y = k x, where k is a constant number called constant of variation.
We have to find the value of k.
y = k x is an equation in slope intercept form where k is the slope and the value of y intercept is 0, since the line passes through origin (0, 0).
It is enough to find the slope of the given line.
Take two points from the graph, let it be (4, 1) and (-4, -1).
Slope = (-1 - 1) / (-4 - 4) = -2 / -8 = 1/4
The value of k is 1/4.
Hence the value of constant of variation is 1/4.
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Why is it desirable to have the explanatory variables spread out to test a hypothesis regarding B1 or construct confidence intervals about B1?
O So the mean of B1 is smaller
O So the mean of B1 is larger
O So the standard deviation of B1 is smaller
O So the standard deviation of B1 is larger
B1 should be smaller in order to spread out the explanatory components in order to test a B1 hypothesis or generate confidence intervals around a B1 hypothesis.
Though an explanatory variable is a hypothesis that is predicted to be able to explain the research outcomes, a response variable demonstrates the impact that is anticipated to come from the explanatory variable. The regression line has a slope that is B1.
In other circumstances, the alternative hypothesis is compared against the null hypothesis to determine whether the assertion that the population slope is not equal to Zero is valid. The value of b1 is to be construed as the average result altering when the explanatory variable is increased by one unit.
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suppose Jana collected another 25/100 of their goal. What fraction of the goal have they now collected?
Answer:
1/4.
Step-by-step explanation:
1/4 is the answer because 25 x 4 = 100. Lets say you had 50/100. That would be equivalent to 2/4 , because 50 x 2 = 100.
25 is often reffered to as a quarter, but if someone says it is lets say a quarter past 5:00, they are trying to say is is 5:15, because 15 x 4 = 60, and 60 minutes is an hour.
nterstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not ). The number density is about and the temperature is about 3 K. What is the edge length L of an LxLxL cube of gas with 1.0 J of thermal energy? I found: P = 4.14x10-17Pa Vrms = 273m/s
Our velocity is said to be 10% of the speed of light. Therefore, we need to multiply by 10% the speed of light. As a result, our speed is three times 10 to 7 m/s. Our diameter is 500 metres.
Thus, our radius is 250 metres. It goes through a volume of space equal to pi r squared times V times T in one second. We then multiply 5.8, 8 times 10 to the 12 m3 by pi times 2, 50 squared times three times 10, and seven times one second. Our density is 10 to 6 atoms per cubic metre, or one atom per cubic centimetre.
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Use the drawing tools to form the correct answer on the provided grid.
During the summer, Krista noted both the number of customers who came to her lemonade stand each day and how much the temperature rose during the day while her stand was open. Based on the data, she concluded that there is a positive correlation between the number of customers and the increase in temperature. Identify which of the two data tables represents a positive correlation. Then, plot the set of data points from that table.
Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.
What is a correlation?
A positive correlation is a relationship between two variables in which an increase in one variable is associated with an increase in the other variable.
Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.
To plot the set of data points from Table 1, we can use a scatter plot. Each data point in the table would be represented by an (x,y) coordinate, where x is the increase in temperature and y is the number of customers. So, in this case, we would have 8 points in the scatter plot, one for each pair of data in table 1.
The plot would look like a group of points in the coordinate plane with a general upward trend, as the x-values (increase in temperature) increase, the y-values (number of customers) also increase. It's possible to observe a pattern in the points, this pattern confirms that there is a positive correlation between temperature and the number of customers.
Hence, Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.
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Which ordered pair is a solution for the system of equations?
3a-5y-15
2-y--4
(2,8)
(-6, -11)
(0, 3)
0 (-5, -6)
The order pair solution of the given system of equations is (-5, -6).
Solving system of equations:Here we will use the substitution method to solve the given equation. In this obtain value of one variable from one equation and substitute the value in another equation to get the value of variables.
Here we have
3x - 5y = 15 ---- (1)
2x - y = - 4 ---- (2)
=> y = 2x + 4 --- (3)
From (1) and (3)
=> 3x - 5(2x + 4) = 15
=> 3x - 10x - 20 = 15
=> -7x = 35
=> x = -5
Substitute x = - 5 in 2x - y = - 4
=> 2(-5) - y = -4
=> -10 - y = -4
=> y = - 6
Therefore,
The order pair solution of the given system of equations is (-5, -6).
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Determine if the sequence below is arithmetic or geometric and determine the common difference / ratio in simplest form. 45,\, 15,\, 5,\, ...
Answer:
The sequence is geometric.
Step-by-step explanation:
Arithmetic Sequence:
A arithmetic sequence is a sequence of numbers in which the next term is calculated by adding some constant amount to the current term. It can be seen almost as a linear equation, but "x" is always a whole number, that starts at zero and increases by one.
We can check if a sequence is arithmetic or not, by subtracting any term from the next term. This amount should be constant for all terms if a sequence is arithmetic. This constant amount can be seen as the slope, how much it's changing by each term.
Geometric Sequence:
A geometric sequence is a sequence of numbers in which the next term is calculated by multiply by some constant amount and the current term. It can be seen almost as a exponential equation, but "x" is always a whole number, that starts at zero and increases by one.
Using the definition above, it can also be thought of as the current term being equal to the previous term multiplied by a constant amount. This means if we divide any term in the sequence by the previous term, this should be a constant amount, no matter which term we use.
Solving the Problem:Now that we know what the definitions of a geometric and arithmetic sequence as well as how to check if a sequence is either, we can now apply this knowledge to the problem. Let's start by checking if the sequence is arithmetic.
We can start by using the first term "45" and subtracting it from the second term "15", which gives us "-30". This means we had to "add" "-30" to get the second term. If this sequence is arithmetic, this means we could add this to the second term and get the third term. If we add "-30" to "15" we get "-15" which is not equal to the next term. So the amount that it's changing by is not constant, meaning this sequence is not arithmetic.
Now let's check if the sequence is geometric. Each term can be defined as the previous term multiplied by some constant amount. So if we divide any term by it's previous term we get this amount that it had to be multiplied by which should be constant. We cannot start with the first term, since there is no previous term before the first term. So let's start with the second term: "15", now let's divide it by the previous term, which is the first term: "45" [tex]\frac{15}{45} = \frac{1}{3}[/tex]. If this sequence is geometric, we can multiply the second term by this 1/3 and get the next term, which is the third term. If we multiply "15" (the second term) by "1/3" we get [tex]\frac{15}{3}[/tex] which is equal to "5", which is equal to the third term. So this sequence appears to be geometric.
(PRE CALC) Which graph shows the function f(x)=−ln(x−1)
Find the absolute value of the fraction. Draw a number line to show how far the fraction is from 0. -5\13
The required number line attached below shows the absolute value of the fraction 5/13 is from 0.
What is absolute value?Absolute value is defined as the magnitude or size of a number, no negatives are allowed. The distance a number is from 0 on the number line is known as the absolute value of the number.
The fraction is given in the question, as follows:
-5/13
We have to determine the absolute value of the fraction.
As per the question, we can write the absolute value of the fraction as:
⇒ 5/13
Thus, the number line attached below displays the distance of the fraction from 0.
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1. The height of a right triangular prism is 1 5/6 inches. Each side of the triangular base measures 10 inches, and the height of the base is 8 2/3 inches. The triangular prism is placed atop a cube whose side measures 10 inches so that one of the triangular prism’s bases lies completely on one side of the cube.
What is the surface area of the solid formed?
100 POINTS AND IF YOU INCLUDE ACCURATE DIAGRAM, BRAINLIEST!!!!!!!
If the height of a right triangular prism is 1 5/6 inches and each side of the triangular base measures 10 inches. The surface area of the solid formed is: 598.33 in².
How to find the Surface area?First step is to convert fraction into simple fraction
Right triangular prism=1 5/6 inches= 11/6 inches
Height of the base=8 2/3 inches = 26/3 inches
Second step is to find the surface area
Surface area = 5(10× 10) + (0.5÷ 10 × 26/3) + 3(10× 11/6)
Surface area = 5(100) +43.33+ 3(18.33)
Surface area = 500+43.33 + 55
Surface area = 598.33 in²
Therefore the surface area of the solid formed is about 598.33 in²
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How do you write 11/15 as a decimal
To write [tex]\dfrac{11}{15}[/tex] as a decimal, you need to divide the numerator by the denominator.
The numerator of the fraction is 11.
The denominator of the fraction is 15.
[tex]11\div15[/tex]
[tex]=\fbox{0.73}[/tex]
please need help!!!!!!!!!!! Answer 8, 9, and 10 please!!!!!!!!!!!! 50 points!!!!!!!!!!!!!!!
Answer:
8) AD = 23
9) m<B = 63
10) G(1,-5)
-------------------------------
8)
AD = BC
Thus, x + 21 = 12x - 1
11x = 22
x = 2
So, AD = 23
9)
consecutive angles are supplementary, sum to 180
y/2 + y - 9 = 180
3/2 y = 189
y = 126
so, m<B = 63
10)
Diagonals bisect each other. The midpoint of DF = (-1.5, 0.5)
-1.5 = 2+x/2
x = 1
0.5 = 6+y/2
y = -5
G(1,-5)
What is the surface area of this
cylinder?
Use ≈ 3.14 and round your answer
to the nearest hundredth.
9 ft
5 ft
The surface area of the cylinder is 791.28 sqft.
What drives your use of surface area?A three-dimensional object's surface area is the sum of all of its faces. Real-world applications of the concept of surface areas include wrapping, painting, and eventually building things to achieve the best possible design.
What is the equation for a hollow cylinder's volume?The formula for the volume of a hollow cylinder is V = (R2 -r2)h cubic units if "R" denotes the outer radius, "r" denotes the inner radius, and "h" denotes the height.
Given,
radius=9ft
height=5ft
[tex]SA=2\pi rh+2\pi r^2\\SA=2*3.14*8*5+2*3.14*5^2\\SA=791.26 sqft[/tex]
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Please help me on this I’m stuck.
The graph of the new trapezoid is attached
What is an transformation?Transformation is the movement of a point in the coordinate plane. Types of transformation are rotation, dilation, reflection and translation.
Dilation is the increase or decrease in size of a figure by a scale factor. The rule for dilation is:
(x, y) ⇒ (kx, ky)
The vertices of the trapezoid is P(1, -2), Q(2, -2), R(2, 1) and S(-2, 1)
If the trapezoid is dilated by a scale factor of 4 about the origin, the new points are:
P'(4, -8), Q'(8, -8), R'(8, 4) and S'(-8, 4)
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Maysol made 75 bracelets this month.
Last month she made 40 bracelets.
What is the percent of change
There is a 46.7% increase in the number of bracelets made.
What is Percentage?A percentage is a number or ratio that can be expressed as a fraction of 100 (100 as denominator). To calculate percent of a number, divide the number by the whole and multiply by 100.
Hence, it means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.
Maysol made;
This month = 75 bracelets
Last month = 40 bracelets
Change = 75 - 40
Change = 35
This means there was an increase
% increase = (35/75) x 100
% increase = 46.7%
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In a bag of marbles, 12% were red, 16% were blue, and the rest are white. If the bag has 250 marbles how many were red or blue
Answer: 70
Step-by-step explanation: First, let's find how many are worth each in numbers. Use this formula:
Part Percent * Whole number / 100
Plug in the numbers:
Red = 12 * 250 / 100
3000/100
= 30 red marbles
Blue = 16* 250 / 100
4000 / 100
= 40 blue marbles
40 + 30 = 70
White = 250 - 70
= 180 white marbles
We can check our answer by adding 12 + 16. That's 28. So by multiplying 250 by 28, we get 7000. 7000 divided by 100 = 70
70 = 70
Therefore, both statements are true
I hope this helped!
ms wong wants to estimate the hourly rate of any given babysitter. how should she estimate the rate? why?
In order to estimate the hourly rate of any given babysitter, Ms. Wong can make use of the mean of the sample, because it most closely describes the values of the data in the dataset
What is the sample mean of a dataset?The sample mean is the average of the values in the data sample.
Possible part of the question, obtained from a similar question online includes the following table of values;
Babysitter [tex]{}[/tex] Rate in dollars per hour
1 [tex]{}[/tex] 9.6
2 [tex]{}[/tex] 9.80
3 [tex]{}[/tex] 15.50
4 [tex]{}[/tex] 10.6
5 [tex]{}[/tex] 9.7
6 [tex]{}[/tex] 10.20
7 [tex]{}[/tex] 8.80
8 [tex]{}[/tex] 11.20
9 [tex]{}[/tex] 8.80
10 [tex]{}[/tex] 10.10
The estimate of the hourly rate of any given babysitter can be found using the descriptive statistic, such as the mean, median, or mode, which are measures that indicate the possible mid value of a data set.
The mean is the average value of the dataset, which is obtained by adding the individual data points together and dividing the sum by the number of data points as follows;
9.6 + 9.8 + 15.5 + 10.6 + 9.7 + 10.2 + 8.8 + 11.2 + 8.8 + 10.1 = 104.3
The number of datapoints in the data = 10
The mean = 104.3/10 = 10.43
The median is the middle value which is found by arranging the values in increasing order, then selecting the middle value as follows;
The median values in the dataset that has 10 data points is the average of the 5th and 6th values, which are; 9.8 and 10.1
The median of the dataset is therefore; (9.8 + 10.1)/2 = 9.95
The mode of the dataset is the most common value within the dataset. The mode is therefore 8.8
Based on the above three values, the best estimate that most closely describe the data in the dataset is the mean
Therefore, to estimate the hourly rate of any babysitter, Ms. Wong can make use of the mean, because it most closely describes the values in the dataset.
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Which list shows the numbers in ascending order? (Least to Greatest)
Responses
|−2.5|, −2.25, 2.75
-2.25,-2.5,2.75 I think
A square based prism with base lengths of 9 has a height of "h" inches. A pyramid with the same base and height is carved out of the prism. What is the volume of the remaining part of the prism?
When the pyramid is cut out of the prism, it has a volume of 36 cubic meters and the same base and height.
what is volume ?It is also known as the object's capacity. The fundamental equation for volume is length, width, and height, as opposed to the fundamental equation for the area of a rectangular shape, which is length, breadth, and height. The math is the same regardless of how you refer to the different dimensions. For example, you can substitute "depth" for "height." Volume is used to describe an object's capacity. Volume can also be used to describe how much space a three-dimensional object occupies.
given
The ratio of their volumes is always one to three
Since the volume of a prism is 108 cubic meters,
a pyramid's volume is 1/3 * 108, or 36 cubic meters.
When the pyramid is cut out of the prism, it has a volume of 36 cubic meters and the same base and height.
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The topic is "Polynomial Synthetic Division". I have no idea how to do it, I need help please :(
Answer:
[tex]x^2+7x-9+\dfrac{12}{x-7}[/tex]
Step-by-step explanation:
The detailed explanation is in the attached image. It is a real pain to type this in the brainly equation editor so I just used LateX and printed out the result in case you are interested.
I may have made typos in the detailed explanation, if so, let me know
Here is a screenshot of my problem:
The approximation of the area of f(x) = x² + 2x, from x = 2 to x = 10, is given as follows:
320 units squared.
How to approximate the area?The area under a curve is approximated using the definite integral of the function that defines the curve within it's bounds.
As it is specified that the area should be approximated using rectangles, a Left Riemann Sum will be used to approximate the area.
First we must obtain the Delta value, which is the difference between the bounds of the integral, divided by the number of intervals, thus, considering four intervals, we have that:
[tex]\Delta_x = \frac{10 - 2}{4} = 2[/tex]
Then the values of x at which the numeric values are calculated are obtained as follows:
[tex]x_i = a + \Delta_x(i - 1)[/tex]
Thus the values of x are of:
[tex]x_1 = 2[/tex][tex]x_2 = 4[/tex][tex]x_3 = 6[/tex][tex]x_4 = 8[/tex]The numeric values are given as follows:
f(2) = 2² + 2(2) = 8.f(4) = 4² + 2(4) = 24.f(6) = 6² + 2(6) = 48.f(8) = 8² + 2(8) = 80.Then the definite integral is obtained as follows:
[tex]\sum_{i = 1}^n \Delta_x f(x_i)[/tex]
Meaning that the result of the integral is of:
2(8 + 24 + 48 + 80) = 320 units squared.
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How many ways can we pay $1.10 with nickel and dime
Answer: We want to know how many ways we can pay $1.10 with nickels and dimes. Since nickels are worth $0.05 and dimes are worth $0.10, we can use different combinations of these coins to make $1.10.
One way to approach this problem is to use the method of generating functions.
Let x be the number of nickels and y be the number of dimes. Then the generating function for this problem is (x+y) = (x+y)(x^0y^0 + x^1y^0 + x^0y^1 + x^2y^0 + x^1y^1 + x^0y^2...)
We know that the sum of x and y should be 11 because 0.05x + 0.1y = 1.10
The coefficient of x^5y^6 in the expansion of (x+y)^11 gives the number of ways to pay $1.10 with 5 nickels and 6 dimes.
The coefficient of x^5y^6 = 252
So there are 252 ways to pay $1.10 with nickel and dime.
Step-by-step explanation:
Find the distance between points G and H
The distance between point G and H is 4√2 units51.96 unit²
What is an equation?An equation is a expression that shows the relationship between numbers and variables.
The distance between two points A(x₁, y₁) and B(x₂, y₂) on the origin is:
[tex]AB=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]
Let us take point G as out reference point (origin)
Hence:
G = (0, 0) and H = (4, -4)
[tex]GH=\sqrt{(-4-0)^2+(4-0)^2}=4\sqrt{2}\ units[/tex]
The distance is 4√2 units
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Dom, the jolly court jester, needs to create a square-based rectangular box with a volume of 32,000 cubic inches for his king at the lowest possible cost. The cost of material for the sides of the box $0.25 per square inch while the cost of material for the top and bottom of the box cost $1.00 per square inch. What are the dimensions of the box that would minimize of the cost? Show all work, please.
Answer:
Dimensions = 20 inches by 20 inches by 80 inches
The floor is 20 inches by 20 inches. The height is 80 inches.
The minimized cost is $2400.
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Work Shown:
h = height of the box, in inches
x = side length of the square base, in inches
x^2 = area of the floor = area of the ceiling
x^2h = volume of the rectangular box
x^2h = 32000
h = 32000/(x^2)
C = total cost in dollars
C = 0.25*(area of the sides) + 1.00*(area of the floor and ceiling)
C = 0.25*(xh+xh+xh+xh) + 1.00*(x^2+x^2)
C = 0.25*4xh + 1.00*2x^2
C = xh + 2x^2
C = x*(32000/(x^2)) + 2x^2
C = (32000/x) + 2x^2
C = (32000/x) + (x*2x^2)/x
C = (32000/x) + (2x^3)/x
C = (32000+2x^3)/x
The goal is to make C the smallest possible, aka we want to minimize it.
Visually we want the lowest point on the cost curve.
We have two options to get this task done:
Use a graphing calculator.Use calculus (specifically derivatives).I'll assume your teacher hasn't gone over calculus at this point. I'll go for option 1 mentioned above.
Use a graphing tool like GeoGebra to plot out the cost function curve. See the diagram below. The lowest point on this curve is at (20,2400). I used the "min" function to determine this lowest point. Keep in mind that x > 0.
This lowest point indicates to us that x = 20 causes C(x) to be the smallest at $2400. This is the minimized cost.
Therefore, the square base should be 20 inches by 20 inches. The height should be:
h = 32000/(x^2) = 32000/(20^2) = 80 inches
The box should be 20 inches by 20 inches by 80 inches.
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Check:
Volume = length*width*height = 20*20*80 = 32000 cubic inches
This helps verify the answer.
Answer:
Width = 20 inches
Length = 20 inches
Height = 80 inches
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6.7 cm}\underline{Volume of a square-based rectangular box}\\\\$V=x^2h$\\\\where:\\\phantom{ww} $\bullet$ $x$ is the side length of the base.\\\phantom{ww} $\bullet$ $h$ is the height.\\\end{minipage}}[/tex]
Given the volume of the square-based rectangular box is 32,000 in³, substitute this into the equation and rearrange to isolate h:
[tex]\implies x^2h=32000[/tex]
[tex]\implies h=\dfrac{32000}{x^2}[/tex]
[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Surface Area of a square-based rectangular box}\\\\$S=2x^2+4xh$\\\\where:\\\phantom{ww} $\bullet$ $x$ is the side length of the base.\\\phantom{ww} $\bullet$ $h$ is the height.\\\end{minipage}}[/tex]
Given:
Cost of material for the sides of the box = $0.25 per in²Cost of material for the top and bottom of the box = $1.00 per in²Create an equation for the total cost, C, of the materials based on the equation for the surface area:
[tex]\implies C=(1)2x^2+(0.25)4xh[/tex]
[tex]\implies C=2x^2+xh[/tex]
Substitute the expression for h into the equation for cost to create an equation for C in terms of x:
[tex]\implies C=2x^2+x\left(\dfrac{32000}{x^2}\right)[/tex]
[tex]\implies C=2x^2+\dfrac{32000}{x}[/tex]
[tex]\implies C=2x^2+32000x^{-1}[/tex]
To find the value of x that would minimize the cost, differentiate the equation for cost:
[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=4x-32000x^{-2}[/tex]
[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=4x-\dfrac{32000}{x^{2}}[/tex]
[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=\dfrac{4x^3-32000}{x^{2}}[/tex]
Set the differentiated equation to zero and solve for x:
[tex]\implies \dfrac{4x^3-32000}{x^{2}}=0[/tex]
[tex]\implies 4x^3-32000=0[/tex]
[tex]\implies 4x^3=32000[/tex]
[tex]\implies x^3=8000[/tex]
[tex]\implies x=20[/tex]
Therefore, the side lengths of the base of the box that would minimize the cost are 20 inches.
To find the height of the box that would minimize the cost, substitute the found value of x into the expression for height:
[tex]\implies h=\dfrac{32000}{20^2}[/tex]
[tex]\implies h=\dfrac{32000}{400}[/tex]
[tex]\implies h=80\; \rm in\;(2\:d.p.)[/tex]
Therefore, the dimensions of the box that would minimize cost are:
width = 20 incheslength = 20 inchesheight = 80 inchesThe teacher gave a true and false quiz where P(true) = 0.5 for each question. Interpret the likelihood that the first question will be true.
Likely
Unlikely
Equally likely and unlikely
This value is not possible to represent probability of a chance event.
The likelihood that the first question will be true is Equally likely and unlikely. That is option C
What is probability?Probability is defined as the possibility of an outcome of an event which may likely or unlikely occur.
The true and false quiz given by the teacher;
P(true) = 0.5 for each question.
Therefore, False = 0.5 for each question making the probability of the first question being true to be equally likely and unlikely.
Learn more about probability here:
https://brainly.com/question/24756209
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