The result of the Integral is t * e^(2t) + C
To evaluate the integral ∫ e^(2t) * 2t * e^t dt, we can use the substitution method.
Let's make the substitution u = e^t. Then, differentiating both sides with respect to t, we get du/dt = e^t.
Rearranging this equation, we have dt = du / e^t.
Now, let's substitute these expressions into the integral:
∫ e^(2t) * 2t * e^t dt = ∫ (2t * e^t) * e^(2t) * (du / e^t)
Simplifying, we have:
∫ 2t * e^(2t) du
Now, we can integrate with respect to u:
∫ 2t * e^(2t) du = t * ∫ 2u e^(2t) du
Integrating, we get:
t * e^(2t) + C,
where C is the constant of integration.
So, the result of the integral is t * e^(2t) + C
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The calculated value of the integral [tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt[/tex] is 0.662
How to evaluate the integralFrom the question, we have the following parameters that can be used in our computation:
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt[/tex]
The above expression can be integrated using integration by substitution method
When integrated, we have
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt = \frac{\ln(e^{2t} + e^{-2t})}{2}|\limits^1_0[/tex]
Expand the integrand for t = 0 and t = 1
So, we have
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt = \frac{\ln(e^{2} + e^{-2})}{2} - \frac{\ln(e^{0} + e^{0})}{2}[/tex]
This gives
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt = \frac{\ln(e^{2} + e^{-2})}{2} - \frac{\ln(1 + 1)}{2}[/tex]
This gives
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt = \frac{\ln(7.524)}{2} - \frac{\ln(2)}{2}[/tex]
Next, we have
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt = 1.009 - 0.347[/tex]
Evaluate the difference
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt = 0.662[/tex]
Hence, the value of the integral is 0.662
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Question
Evaluate the integral.
[tex]\int\limits^1_0 {\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}} \, dt[/tex]
A recipe for dessert calls for 1/3 cup of powdered sugar and 2/6 cup of brown
sugar. What is the total amount of sugar needed for the recipe
Answer:
2/3
Step-by-step explanation:
1/3 cup powdered sugar
2/6 cup brown sugar
Reduce 2/6 to 1/3.
1/3 + 1/3 = 2/3
Evaluate the function when x=5
F(5)=?
Step-by-step explanation:
When evaluating piecewise function, make sure the x value satisfies the function we use.
since 5>3, we use the second equations
That function is a constant function, this means at any x>3, exclusive, we will have a y value of -1.
So
[tex]f(5) = - 1[/tex]
Can someone please tell me which one is correct??? The first screenshot is the question and the second screenshot is the options I have. I am in dire need of help
Answer:
43.5 unit^2.
Step-by-step explanation:
Area of the triangle
= 1/2 * base * height
= 1/2 * AB * AC.
AB = √ [(6-1)^2 + (6-8)^2] = √29
AC = √ [(8 - (-7))^2 + (1 - (-5)^2] = √261
So the area = 1/2 * √29 * √261
= 43.5
Which phrase describes an unknown or changeable quantity?
Answer:
That is a variable.
Step-by-step explanation:
- opposed to a constant which is known and does not change.
Jacob is cutting a tile in the shape of a parallelogram. Two opposite angles have measures of (6n − 70)° and (2n + 10)°.
What are the two different angle measures of the parallelogram-shaped tile?
20° and 160°
50° and 130°
30° and 150°
70° and 110°
A parallelogram is a quadrilateral in which opposite sides are equal, and the opposite angles sum up [tex]180^{o}[/tex]. Thus, the measures of the two different angles in the question are 70° and 110° i.e option D.
A parallelogram is a quadrilateral in which opposite sides are equal, and the opposite angles sum up [tex]180^{o}[/tex]. It has four straight sides and can be classified as a quadrilateral.
Thus from the given question, we have:
(6n − 70)° + (2n + 10)° = [tex]180^{o}[/tex]
8n - 60 = [tex]180^{o}[/tex]
8n = [tex]180^{o}[/tex] + 60
8n = 240
n = [tex]\frac{240}{8}[/tex]
= [tex]30^{o}[/tex]
n = [tex]30^{o}[/tex]
So that,
i. (6n − 70)° = (6[30] − 70)°
= 180 - 70
= [tex]110^{o}[/tex]
ii. (2n + 10)° = (2[30] + 10)°
= 60 + 10
= [tex]70^{o}[/tex]
Therefore, the measures of the two different angles are 70° and 110°. Option D.
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Answer:
b, 50° and 130°
Step-by-step explanation:
edge 2023 :)
^ that shi sucks btw we needa boycott the program or smt
Change the equation of the parent square root function to represent the equation of the graphed function. Enter the correct answer in the box.
The equation of the parent square root function to represent the equation of the graphed function will be, y=√x-2
According to the statement
we have to show the square root function as a equation in the graphical representation.
So,
we know that the definition of a
Graph a diagram showing the relation between variable quantities, typically of two variables and it also show the relation between more than two variables.
Now, we know that the definition of Equation is a mathematical statement consisting of an equal symbol between two algebraic expressions with the same value is known as an equation.
And the equation obtained from the graph is a y=√x-2 by a some calculations in the graph.
So, The equation of the parent square root function to represent the equation of the graphed function will be, y=√x-2
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What is the value of f(16) - f(0) when f(x) = 4x - 8?
16
48
56
64
Answer:
64
Step-by-step explanation:
evaluate by substituting x = 16 and x = 0 into f(x)
f(16) = 4(16) - 8 = 64 - 8 = 56
f(0) = 4(0) - 8 = 0 - 8 = - 8
then
f(16) - f(0) = 56 - (- 8) = 56 + 8 = 64
Answer:
64
Step-by-step explanation:
f(x) = 4x - 8
First find f(16)
f(16) = 4(16) -8 =64-8=56
Then find f(0) = 4)0) -8 = -8
f(16) - f(0) = 56 - (-8) = 56+8 = 64
Fill in the blank: In data analytics, a _____ refers to all possible data values in a certain dataset
In data analytics, a population refers to all possible data values in a certain dataset
Data analysis is a systematic computer-aided analysis of data or statistics. [1] It is used to discover, interpret, and convey meaningful patterns in the data. It also includes applying data patterns for effective decision-making.
It may be useful in areas where there is a lot of recorded information. The analysis relies on the simultaneous application of statistics, computer programming, and operations research to quantify performance.
Organizations can apply analytics to business data to describe, predict, and improve business performance. Areas within the analysis include descriptive analysis, diagnostic analysis, predictive analysis, prescriptive analysis, and cognitive analysis in particular. [2] Analysis can be applied in various fields such as marketing, management, finance, online systems, information security, and software services.
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Twenty-one is 25% of what number?
A. 5.25
B. 84
C. 10.5
D. 63
Answer: 84
Step-by-step explanation:
When substituting this question into an algebraic equation you can write it as:
21=.25x
(x clearly represents the number you are trying to find)
When you divide both sides by .25 to isolate x you get x=84
Shawna, Ryan, and Dominic ate a whole pizza. Shawna ate 1
3 of the pizza
and Ryan ate 1
2 of the pizza. Dominic ate the rest of the pizza. How much
pizza did Dominic eat?
Answer:
[tex]\frac{1}{6}[/tex]
Step-by-step explanation:
You need to add the amount Shawna & Ryan ate together, then subtract it from 1 to find the amount of pizza that was left for Dominic to eat.
[tex]\frac{1}{3}[/tex] + [tex]\frac{1}{2}[/tex] = ?
Make all the denominators the same to add fractions.
[tex]\frac{1}{3}[/tex] is [tex]\frac{2}{6}[/tex]
[tex]\frac{1}{2}[/tex] is [tex]\frac{3}{6}[/tex]
[tex]\frac{2}{6}[/tex] + [tex]\frac{3}{6}[/tex] = [tex]\frac{5}{6}[/tex]
1 - [tex]\frac{5}{6}[/tex] = [tex]\frac{1}{6}[/tex]
A cross section is made by the intersection of a plane and a square pyramid at an angle either parallel or perpendicular to the base. the cross section can be which of these shapes? select three options.
A cross-section is made by the intersection of a plane and a square pyramid at an angle either parallel or perpendicular to the base square, triangle, and, trapezoid.
What is an area of cross-section?A cross-section parallel to the base will be a square.
One perpendicular to the base will be a trapezoid, or if it passes through the vertex of the pyramid, a triangle.
The cross-section of the pyramid is perpendicular to the base and through the vertex will be a triangle.The cross-section has the same shape but it has a smaller base.The cross-section perpendicular to the triangular bottom then it will be a trapezoid.Hence, a cross-section is made by the intersection of a plane and a square pyramid at an angle either parallel or perpendicular to the base square, triangle, and, trapezoid.
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A rose garden is formed by joining a rectangle and a semicircle, as shown below. The rectangle is 29 ft long and 20 ft wide.
Find the area of the garden. Use the value 3.14 for , and do not round your answer. Be sure to include the correct unit in your answer.
Answer:
15.7 square feet
Step-by-step explanation:
To find the area of the garden we find the area of the rectangle and the area of the semicircle.
The area of the rectangle:
A=bh
A=(29)(20)
A=580ft^2
As shown in the picture, the width of the rectangle is 20 ft wide, which means that the diameter of the semicircle is also 20ft wide. The diameter is two times the radius, which means the radius of the semicircle is 10ft.
The area of a circle is represented by the equation: [tex]A=\pi r^{2}[/tex]
A semicircle is half of a circle, therefore the equation to find the area of a semicircle is: [tex]A=\frac{1}{2}\pi r^{2}[/tex]
Plugging our radius of 10ft in we get:
[tex]A=\frac{1}{2}\pi (10)\\ A=5\pi[/tex]
We use 3.14 for our value of [tex]\pi[/tex] to get:
[tex]A=5(3.14)\\A=15.7ft^{2}[/tex]
Therefore, the area of the rose garden is 15.7 square feet.
2. According to Archimedes, the area of any circle is equal to the area of a right triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Verify this formula for yourself using the formula for the circumference of a circle.
Formula [tex]\text { area }=1 / 2 \times \text { base } \times \text { height }=1 / 2 \times 2 \pi r \times r=\pi r^{2}[/tex] for the circumference of a circle.
How did Archimedes find the circumference of a circle?Archimedes stated in his Proposition that the area of a circle is equal to the area of a triangle with a base equal to the circumference and a height equal to the radius: (1/2)(r · 2πr) = πr2. Archimedes arrived at his approximation of the circumference of the circle by increasing the number of sides of the hexagon. Archimedes claimed that the area of any circle is equal to the area of a right triangle, where the radius of the circle is represented by one side and the circumference by the other.
Archimedes demonstrated using a similar method that the area of a circle of diameter D is equal to the area of a right-angled triangle with one side equal to the radius and the other to the circumference of the circle on the right angle.
Formula for the circumference of a circle:
A circle's area is equal to pi times the radius squared (A = r2). Discover how to apply this formula to determine a circle's area given its diameter.
The circumference is diameter x pi, or 2 x radius x pi.
[tex]\text { area }=1 / 2 \times \text { base } \times \text { height }=1 / 2 \times 2 \pi r \times r=\pi r^{2}[/tex].
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Which of these is the equation of a graph
in which the vertex is (4, 2) and
a=-2
HELPPPPPPPP
The equation of the graph in which the vertex (4,2) and a=-2 is
y=-2[tex](x-4)^{2}[/tex]+2.
Given that the point of the vertex is (4,2) and a=-2.
Equation is like a relationship between two or more variables expressed in equal to form. Equations of two variables look like ax+by=c. It may be a linear equation,quadratic equation, cubic equation or many more depending on the power of variable in the equation.
We know that point of vertex look like (h,k) in the equation and equation look like as under:
y=a[tex](x-h)^{2} +k[/tex].-----------1
So in order to find the equation we have to just put h=4 and k=2 and a=2 in equation 1.
y=-2[tex](x-4)^{2} +2[/tex]
Hence the equation of the graph in which the vertex (4,2) and a=-2 is
y=-2[tex](x-4)^{2}[/tex]+2.
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The lateral surface area of cone A is exactly 1/2 the lateral surface area of cylinder B. Cone A radius is r and height h - Cylinder B radius is r and height h. True or false?
The ratio of the lateral surface area of cone A to the lateral surface area of cylinder B is equal to [tex]r = \frac{1}{2}\cdot \frac{\sqrt{r^{2}+h^{2}}}{h}[/tex]. (Correct choice: False)
What is the ratio of lateral area of cone to the lateral area of the cylinder?In accordance with space geometry, the lateral areas of the cone and cylinder are described by the following equations:
Cone
[tex]A_{l} = \pi \cdot r \cdot \sqrt{r^{2}+h^{2}}[/tex] (1)
Cylinder
[tex]A_{l} = 2\pi\cdot r\cdot h[/tex] (2)
If we divide (2) by (1), then we have the following ratio:
[tex]r = \frac{1}{2}\cdot \frac{\sqrt{r^{2}+h^{2}}}{h}[/tex]
The ratio of the lateral surface area of cone A to the lateral surface area of cylinder B is equal to [tex]r = \frac{1}{2}\cdot \frac{\sqrt{r^{2}+h^{2}}}{h}[/tex]. (Correct choice: False)
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A region in the first quadrant is bounded by the graph y equals two thirds times x, the y-axis, and the horizontal line y
By definite integrals and area formula of a rectangle, we find that the area of the region in the first quadrant is 147 / 4 square units.
How to calculate the area of the region by definite integrals
Integrals can be used to determine the area of regions bounded by curves set on Cartesian plane. The upper limit of the integral of the question is initially found:
(2 / 3) · x = 7
x = 21 / 2
The area can be defined as an rectangle in the first quadrant minus the area below the linear equation:
[tex]A = \left(\frac{21}{2} \right) \cdot (7) - \frac{2}{3} \int\limits^{\frac{21}{2} }_{0} {x} \, dx[/tex]
[tex]A = \frac{147}{2} - \frac{1}{3} \cdot \left[\left(\frac{21}{2} \right)^{2}-0^{2}\right][/tex]
A = 147/4
By definite integrals and area formula of a rectangle, we find that the area of the region in the first quadrant is 147 / 4 square units.
Remark
The statement is poorly formatted and incomplete. Correct form is shown below:
A region in the first quadrant is bounded by the line y = (2/3) · x, the y-axis and the horizontal line y = 7. Determine the area of the region.
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A teacher wants to know whether their course helps students on the SAT. The hypothesized population mean for SAT scores is 500. The standard deviation of the population is 77. The sample size is 100. The sample mean for students who took the course is 533. What is the z-score
The z-score is -0.428
What is a z-score?A z-score, also known as a standard score, provides information on how far a data point is from the mean. Technically speaking, however, it's a measurement of how many standard deviations a raw score is from or above the population mean.
You can plot a z-score on a normal distribution curve.
You must be aware of the mean and population standard deviation to use a z-score.
Z-scores allow results to be compared to a "normal" population.
According to the question,
x=500
μ=533
σ=77
z-score=(x-μ)/σ
On substituting the values,
z-score=(500-533)/77
= -0.428
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If sin A + sin^3 A = cos^2 A, prove that cos^6 A - 4cos^4 A + 8cos^2 A = 4.
Please provide all steps. Thank You.
Answer:
Step-by-step explanation:
sinA(1+sin^2A) = cos^2A
sinA(2 -cos^2A) = cos^2A
Squaring both sides,
sin^2A(4-4cos^2A +cos^4A) = cos^4A
(1-cos^2A)(4-4cos^2A +cos^4A) = cos^4A
4-4cos^2A +cos^4A-4cos^2A+4cos^4A-cos^6A = cos^4A
4 -cos^6A +4cos^4A -8cos^2A = 0
cos^6A - 4 cos^4A + 8cos^2A = 4
hence proveproven
Answer:
Step-by-step explanation:
sinA+sin
3
A=cos
2
A
⇒sinA[1+sin
2
A]=cos
2
A
⇒sinA[1+1−cos
2
A]=cos
2
A
squaring on both sides.
⇒sin
2
A[4+cos
4
A−4cos
2
A]=cos
4
A
⇒(1−cos
2
A)[4+cos
4
A−9cos
2
A]=cos
4
A
⇒4+cos
4
A−4cos
2
A−64cot
2
A−cos
6
A+9cos
4
A=cos
4
A
⇒
cos
6
A−4cos
4
A+8cos
2
A=4
Hence Prove
find the midpoint of (0, 4) and (2, -2)
Answer:
[tex]\text{Midpoint} = (1, \; 1)[/tex]
Step-by-step explanation:
[tex]M = (x_M, \; y_M)[/tex]
[tex]M = \left(\dfrac{x_1 + x_2}{2}, \; \dfrac{y_1 + y_2}{2}\right)[/tex]
[tex]M = \left(\dfrac{0 + 2}{2}, \; \dfrac{4 + -2}{2}\right)[/tex]
[tex]M = \left(\dfrac{2}{2}, \; \dfrac{2}{2}\right)[/tex]
[tex]M = (1, \; 1)[/tex]
Answer: (1, 1)
Step by Step Explanation:
The midpoint formula is (x + x/2 , y + y/2)
If you plug in the coordinates it looks like this
( 0 + 2/2 , 4 + (-2)/2)
That turns into this:
(2/2 , 2/2)
Which gives you the final answer of (1, 1)
which tree diagram shows all the possible outcomes for 2 coin flips?
Answer:
c
Step-by-step explanation:
im smart like that
Which transformation(s) can be used to map one triangle onto the other? select two options. reflection only translation only dilation, then translation rotation, then translation rotation then dilation
Transformation which can be used for mapping of one triangle to another are;
Only translationRotation then dilation.What does mapping mean in geometry?
Any method that is outlined for assigning a specific object from one (or the same) set to each object in another is known as mapping. Any set, including all whole integers, all the points on a line, or all the objects inside a circle, can be mapped.We need to map one triangle to another ;
Hence, for mapping of one triangle to another,
The transformation which can be used from the given options are;
Only translationRotation then dilationLearn more about mapping
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Answer:
B. translation only
E. rotation then dilation
Step-by-step explanation:
B. translation only
E. rotation then dilation
Which one of these is not a step used when constructing an inscribed square using
technology? (5 points)
1) create a circle using the center with given point tool.
2) connect the point with a line through the center of the circle.
3) mark the points of intersection between the three circles.
4) create another circle with the same radius as the original.
The correct option is option () Connect the point with a line through the center of the circle.
According to the problem,
a round planar figure whose circumference is made up of points that are equally spaced from a fixed point (the centre).referring to a unit of measurement whose area is equal to a square whose side is the specified unit.A figure that is inscribed in a circle has vertices that are points on the circle. Learn how to draw equilateral triangles, squares, and regular hexagons that are inscribed in circles, as well as how to build figures in circles.So, the correct option is option ()Connect the point with a line through the center of the circle.
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I really need help with this question
Answer:
39
Step-by-step explanation:
I assume the father's age is a 2-digit number. He is not 9 years old or younger, and he is not 100 years old or older.
The sum of the digits of a two-digit number goes up by 1 from one number to the next unless the number ends in 9.
For example, from 17 to 18, the sum of the digits goes from 8 to 9.
If the number ends in 9, for example, 29, the digits add to 11. Then the next number is 30, and now the sum of the digits is 3, which is less than 11.
The father's age now is a number whose sum of digits is not only greater than the sum of the digits next year, but it is 3 times greater.
Let's look at 2-digit number and the next number and the sum of their digits.
19, 20; sums: 10, 2; 10/2 = 5, not 3
29, 30; sums: 11, 3; 11/3 ≠ 3
39, 40; sums: 12, 4; 12/4 = 3
Answer: The father is 39 years old.
Answer: 39 years
Step-by-step explanation:
The father says that adding both of the digits in his age will be 3 times greater than adding both digits one year later. Before we even start answering the problem, this tells us that the father's age has 2 digits in it.
Let's represent the 10's digit with x, and the 1's digit with y. If the father's age was 45, for example, x would be 4 and y would be 5.
Let's first make an equation to represent this situation:
The sum of the digits of the father's age would be represented by x + y, as x and y are the 10's digit and 1's digit respectively.
The father's age next year would be one greater than the current year, which makes y become y + 1. Then, the sum of the digits would be x + y + 1.
Now, we must make the first equation 3 times larger than the second and solve for the sum.
[tex]x + y = 3(x+y+1)\\ x+y=3(x+y)+3\\ 1(x+y)-3(x+y)=3\\ -2(x+y)=3\\ x+y=\frac{3}{-2}=-\frac{3}{2}[/tex]
However, we can neither have a fraction or a negative as a sum of two digits in an age, as digits are always positive integers from 0 to 9.
This flaw came from the assumption that after 1 year, the sum of the digits will be x + y + 1. This isn't always true, as any number with its 1's digit of 9 turns into 0 the next year, and not 10. Instead, the x value increases by 1.
For example, someone the age 69 becomes 70 the next year, x increases by 1 and y becomes 0.
Let's make a new equation to try and see this.
We know that his current age has to have y = 9 for his new age to be y = 0. Therefore, the current sum of digits is x + 9.
The next year, we know that the 10's place will increase by 1 and the 1's place becomes 0. Therefore the sum of digits is x + 1 + 0, or x + 1.
Since the current sum of digits is 3 times greater than the sum next year, let's multiply the second equation by 3 to get equality. We just need to calculate the 10's digit, as the 1's digit is already 9.
[tex]x+9 = 3(x+1)\\ x+9=3x+3\\ 9=2x+3\\ 6=2x\\ x=3[/tex]
Since the 10's digit (or x) is 3 and the 1's digit (or y) is 9, the father's age is 39, and he will turn 40 next year.
Kylie is writing a persuasive essay for English class. The essay has to present her opinion and evidence for constructing that opinion. The essay has to have a minimum word count of 150 words and a maximum word count of 500 words.
Which compound inequality represents the length, w, of the essay?
500 ≤ w ≤ 150
150 ≥ w
500 ≤ w
150 ≤ w ≤ 500
Answer:
The last answer is correct.
Step-by-step explanation:
The w stands for the number of words that she can have. She has to have at least 150 words so the w has to be equal to 150 or greater. The w also has to be less then or equal to 500
Solve the triangle. Round your answers to the nearest tenth.
The missing information of the triangle is C = 63°, b ≈ 28.084 and c ≈ 25.084. (Correct choice: C)
How to determine the missing sides and angles
In this question we have a triangle with a known side lengths and two known angle measures. First, we find the missing angle by Euclidean geometry:
C = 180° - 94° - 23°
C = 63°
Lastly, we determine the missing sides by law of sines:
[tex]b = 11 \times \frac{\sin 94^{\circ}}{\sin 23^{\circ}}[/tex]
b ≈ 28.084
[tex]c = 11 \times \frac{\sin 63^{\circ}}{\sin 23^{\circ}}[/tex]
c ≈ 25.084
The missing information of the triangle is C = 63°, b ≈ 28.084 and c ≈ 25.084. (Correct choice: C)
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Which system of inequalities is shown?
O A. y
y< 4
) B. y>x
y > 4
O C. y>x
y < 4
y > 4
Answer:
y < x
y < 4
Step-by-step explanation:
Dotted line means < or >, and since the shaded region is below the lines y = 4 and y = x, the answer is A.
Very much appreciated!!!
Answer:
80
Step-by-step explanation:
The sum of the interior angles of a triangle is 180. Subtract out the 2 angles that you know.
180 - 70 - 30 = 80
A rectangle has a side length of two and seven eighths feet and a side width of 5 feet. What is the area of the rectangle?
A. six and five eighths ft2
B. nine and two eighths ft2
C. fourteen and one eighths ft2
D. fourteen and two eighths ft2
Answer:
14 3/8 ft^2
Step-by-step explanation:
Area = length * width
= 2 7/8 * 5
= 23/8 * 5
= 115/8
= 14 3/8 ft^2.
The vertices of a rectangle are Q (2, -3), R (2,4), S (5,4), and T (5, -3).
4. Translate rectangle QRST 3 units left and 3 units down to produce rectangle Q'R'S'T'. Find the area of QRST and the area of rectangle Q'R'S'T'.
5. Compare the areas. Make a conjecture about the areas of a preimage and its image after a
translation.
6. The vector PQ=(4,1) describes the translation of A (-1, w) onto A' (2x+1, 4) and B (8y-1,
1) onto B' (3,32). Find the values of w, x, y, and z.
4) [tex]A_{QRST} = A_{Q'R'S'T'} = 9[/tex].
5) The areas of the rectangles QRST and Q'R'S'T' are the same since the translation of the geometric loci conserved the Euclidean distance.
6) x = 1, w = 3, y = - 1/4, z = 1/3
How to analyze and apply rigid transformations
Rigid transformations are transformations applied on geometric loci such that Euclidean distance is conserved. In this question we have applications of translations, a kind of rigid transformation.
Exercise 4
In this part we must determine the areas of rectangles QRST and Q'R'S'T':
Rectangle QRST
A = RS · QT
A = 3 · 3
A = 9
Rectangle Q'R'S'T'
Q'(x, y) = (2, - 3) + (- 3, - 3)
Q'(x, y) = (- 1, - 6)
R'(x, y) = (2, 4) + (- 3, - 3)
R'(x, y) = (- 1, 1)
S'(x, y) = (5, 4) + (- 3, - 3)
S'(x, y) = (2, 1)
T'(x, y) = (5, - 3) + (- 3, - 3)
T'(x, y) = (2, - 6)
A = R'S' · Q'T'
A = 3 · 3
A = 9
The rectangles QRST and Q'R'S'T' have both an area of 9 square units.
Exercise 5
The areas of the rectangles QRST and Q'R'S'T' are the same since the translation of the geometric loci conserved the Euclidean distance.
Exercise 6
In this case, we must solve the following equations:
PQ = A'(x, y) - A(x, y)
(4, 1) = (2 · x + 1, 4) - (- 1, w)
(4, 1) = (2 · x + 2, 4 - w)
(4, 1) = (2 · x, - w) + (2, 4)
(2, - 3) = (2 · x, - w)
(x, w) = (1, 3)
PQ = B'(x, y) - B(x, y)
(4, 1) = (3, 3 · z) - (8 · y - 1, 1)
(4, 1) = (2 - 8 · y, 3 · z)
(4, 1) = (- 8 · y, 3 · z) + (2, 0)
(2, 1) = (- 8 · y, 3 · z)
(y, z) = (- 1/4, 1/3)
To learn more on translations: https://brainly.com/question/17485121
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pls help ILL MARK BRAINLIEST
Answer:
a) Height of the ledge is 1.6 m.
b) roots of the equation are: [tex]\bf d = 2[/tex] , [tex]\bf d = -1[/tex]
c) Maximum height reached = 1.8 m.
Step-by-step explanation:
a) The height of the ledge is the same as the height at which Jake is when he hasn't moved any horizontal distance from the ledge yet, that is, when d = 0:
[tex]h = -0.8(0)^2 + 0.8(0) + 1.6[/tex]
⇒ [tex]\bf h = 1.6 \space\ m[/tex]
∴ Height of the ledge is 1.6 m.
b) The x-intercepts occur where y = 0, that is when h = 0:
[tex]-0.8d^2 + 0.8d + 1.6 = 0[/tex]
Divide both sides of the equation by -0.8:
[tex]d^2 - d -2 = 0[/tex]
Factorizing:
⇒ [tex]d^2 -2d + d - 2 = 0[/tex]
⇒ [tex]d(d-2) +1 (d-2) = 0[/tex]
⇒ [tex](d-2)(d+1) = 0[/tex]
⇒ [tex]d - 2 = 0[/tex] or [tex]d + 1 = 0[/tex]
∴ roots are : [tex]\bf d = 2[/tex] , [tex]\bf d = -1[/tex]
c) To find the maximum value of a quadratic equation in the form
y = ax² + bx + c , use the formula:
max = c - (b² / 4a).
Using the formula for [tex]h = -0.8x^2 + 0.8x + 1.6[/tex] :
max h = [tex]1.6 - ( \frac{0.8^2}{4(-0.8)} )[/tex]
= 1.8
∴ Maximum height reached = 1.8 m.
Answer:
a) 1.6 m
b) -1, 2 meters
c) 1.8 m
Step-by-step explanation:
Apparently, technology tools are allowed when solving this problem. They readily show you the x- and y-intercepts and the vertex of the graph.
a) ledge heightThe height of the ledge is the value of h when d=0. It is the constant in the given equation, and the y-intercept of the graph.
The height of the ledge is 1.6 meters.
b) rootsThe x-intercepts are the values of d that make h equal to zero. The graph shows them to be ...
d = -1
d = 2 . . . . meters
c) maximum heightThe maximum height is the "h" coordinate of the vertex (d, h).
The maximum height is 1.8 meters.
__
Additional comment
"x" is the generic independent variable. The horizontal axis of a graph is often called the "x-axis" and places where the graph crosses that axis are called "x-intercepts" even when the independent variable is named something else. Here, the independent variable is "d", not "x".
Similarly, "y" is the generic dependent variable, and the vertical axis of a graph is often called the "y-axis" even when the dependent variable is something else. This is why we refer to h when d=0 as the "y-intercept". It is the point where the graph crosses the vertical axis.