The evaluation of the triple integral ∫∫∫E 4[tex]x^{2}[/tex] + 3dV is (38/15)ππ
To evaluate the triple integral ∫∫∫E 4x^2 + 3dV in spherical coordinates, we need to express the integrand and the volume element dV in terms of the spherical coordinates ρ, θ, and φ.
The volume element dV in spherical coordinates is given by:
dV = sin φ dρ dθ dφ
where ρ is the radial distance, θ is the azimuthal angle, and φ is the polar angle.
The region E in which we are integrating can be defined in spherical coordinates as follows:
0 ≤ ρ ≤ 2
0 ≤ θ ≤ 2π
0 ≤ φ ≤ π/2
Substituting these expressions into the volume element, we have:
dV = sin φ dρ dθ dφ
= (sin φ) dρ dθ dφ
Now, we need to express the integrand 4[tex]x^2[/tex] + 3 in terms of the spherical coordinates.
The variable x can be expressed in terms of the spherical coordinates as:
x = ρ sin φ cos θ
Therefore, 4[tex]x^2[/tex] + 3 can be expressed as:
4[tex]x^2[/tex] + 3 = 4 [tex]sin^2[/tex] φ [tex]cos^2[/tex] θ + 3
Substituting this expression into the triple integral, we have:
∫∫∫E 4[tex]x^2[/tex] + 3dV
Now, we can evaluate the integral by performing the integration in the order φ, θ, ρ.
= (8/15)π + 2π
= (38/15)ππ
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. Select two choices that are true about the function f(x)
A There is an asymptote at x = 0.
☐ B There is a zero at 23.
OC
There is a zero at 0.
D
There is an asymptote at y = 23.
23x+14
x
Answer:
A. There is an asymptote at x = 0.
D. There is an asymptote at y = 23.
Convert the number 35/4 into decimal form rounded to the nearest hundred.
Answer: 8.75
Step-by-step explanation:
We know that 4 goes into 35 eight times.
35 - (4 * 8) = 3
Next, we know that 3/4 is equal to 0.75 by dividing.
This leaves us with 8.75. Eight wholes and a part of 0.75.
(Dilations MC)
Triangle ABC with vertices at A(-3, -3), B(3, 3), C(0, 3) is dilated to create triangle A'B'C' with vertices at A(-6, -6), B(6, 6), C(0, 6). Determine the scale factor used.
02
1|2
03
-in
The scale factor used is 2.
What is triangle?It is one of the simplest polygon shapes and is commonly used in mathematics and geometry. The sum of the internal angles of a triangle is always 180 degrees.
Define vertices of triangle?The vertices of a triangle are the three points in a two-dimensional (2D) or three-dimensional (3D) space that define the corners or corners of the triangle. In a 2D plane, the vertices are typically denoted as A, B, and C, and in a 3D space, they can be represented as (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) where (x, y, z) are the coordinates of each vertex along the x, y, and z axes respectively. The vertices of a triangle are connected by three line segments, known as edges, to form the sides of the triangle. The combination of the three vertices and the edges connecting them determines the shape and size of the triangle.
To find the scale factor used to dilate triangle ABC to A'B'C', we can compare the corresponding side lengths of the two triangles.
The distance between A(-3, -3) and B(3, 3) is √((3-(-3))^2 + (3-(-3))^2) = 6√2.
The distance between A'(-6, -6) and B'(6, 6) is √((6-(-6))^2 + (6-(-6))^2) = 12√2.
So the scale factor used to dilate triangle ABC to A'B'C' is:
scale factor = length of corresponding side in A'B'C' / length of corresponding side in ABC
= (12√2) / (6√2)
= 2
Therefore, the scale factor used is 2.
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The circumstances if the base of the cone is 12π cm. If the volume of the cone is 96π, what is the height
pleaseee helppp!!
Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.
what is volume ?A three-dimensional object's volume is a measurement of how much space it takes up. It is a real-world physical number that can be expressed in cubic measurements like cubic metres (m3), cubic centimetres (cm3), or cubic feet (ft3). Physics, chemistry, architecture, and mathematics all use the idea of volume extensively. Volume is frequently used to refer to the amount of space that an object or substance takes up, for instance the amount of a container, the volume of either a liquid, or the quantity of a gas. Depending on an object's shape, a different formula is required to determine its volume.
given
The formula V = (1/3)r2h, where V is the volume, r is the radius of the base, and h is the height, can be used to determine the volume of a cone.
Hence, by multiplying the circumference by two, we can determine the radius of the base:
12π / 2π = 6
Thus, the base's radius is 6 cm.
Also, we are informed that the cone's volume is 96. As a result, we can get the height using the following formula for a cone's volume:
V = (1/3)r2h
96 = (1/3)(6/2)h
96 = 36 h
96 / 36 = 8/3
Hence, the cone is 8/3 cm tall as we can get the height using the following formula for a cone's volume.
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PLS HELP ASAP 50 POINTS AND BRAINLEIST!!!
Explain how the arc BX, central angle BCX, and inscribed BNX are connected. what are the relationships between them?
Answer:
See explanation.
Step-by-step explanation:
The arc BX, central angle BCX, and inscribed angle BNX are connected through the following relationships:
1. The central angle BCX subtends the arc BX. This means that the central angle is formed by two radii connecting the center of the circle to the endpoints of the arc BX.
2. The inscribed angle BNX subtends the same arc BX. This means that the inscribed angle is formed by two chords connecting a point on the circumference of the circle to the endpoints of the arc BX.
3. The relationship between the central angle BCX and the inscribed angle BNX is given by the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half the measure of the central angle subtending the same arc. In other words, if θ is the measure of the central angle BCX and α is the measure of the inscribed angle BNX, then:
[tex]\alpha =\frac{1}{2}[/tex] θ
The city is 30 miles long and two-thirds as wide, and 555,000 citizens currently live there. The mayor calculates that the minimum number of people who would have to move outside the city for adequate services to be maintained is 75,000. Enter the maximum population density , in citizens per square mile , that is assumed in the mayor's calculation
The maximum population density evaluated is 1200 citizens per square mile, under the condition that the city is 30 miles long and two-thirds as wide, and 555,000 citizens currently live there.
Now to evaluate the maximum population density that is considered in the mayor's calculation is
Let us first calculate the area of the city which is (2/3) × (30 miles)
= 20 miles.
So, now we can calculate the current population density which is
555,000 / (20 × 20)
= 1387.5 citizens per square mile.
Hence the mayor evaluates that at least 75,000 people must transfer out of the city for adequate services to be exercised, we can find the new population as
555,000 - 75,000
= 480,000 citizens.
Therefore, the new population density would be 480,000 / (20 × 20)
= 1200 citizens per square mile
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The probability of an event is given. Find the odds in favor of the event.
0. 5
The odds in favor of the event are 1.
The probability of an event is the ratio between the total number of favorable outcomes and the total number of outcomes.
The odds in favor of an event are the ratio of the total number of favorable outcomes and the total number of unfavorable outcomes. If the probability of the event is given, we can find the odds in favor by using the formula of odds in favor:
odds in favor = probability of event/probability of not event
In this case question, the probability of an event is given as 0.5 which means that the total number of favorable outcomes is 50 out of 100
So the probability of not having an event is also 0.5 (the other half of the part.)
So, odds in favor = 0.5/0.5 = 1
The probability of the happening of an event is the same as the probability of not happening of an event. This means that the odds in favor of the event are 1 to 1, or simply 1.
Therefore, the odds in favor of the event are 1.
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Guided practice
it's not letter c. -2.6
state what number you would subtract from each side of the inequality to solve the inequality.
5.7 ≥ k + 3.1
a.
3.1
b.
5.7
c.
–2.6
The value of k that is less than or equal to 2.6
To solve the inequality 5.7 ≥ k + 3.1, you should subtract 3.1 from each side of the inequality.
To isolate the variable k, we need to perform the same operation on both sides of the inequality. In this case, we need to subtract 3.1 from each side:
5.7 - 3.1 ≥ k + 3.1 - 3.1
This simplifies to:
2.6 ≥ k
Therefore, the correct answer is:
k ≤ 2.6
We subtracted 3.1 from each side to isolate the variable k, resulting in the inequality k ≤ 2.6. This means that any value of k that is less than or equal to 2.6 will satisfy the original inequality 5.7 ≥ k + 3.1.
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(8-6b)(5-3b)=
You have to find the product this is geometry
The product of (8-6b)(5-3b), using the distributive property of multiplication is [tex]18b^2 - 54b + 40[/tex].
This problem is actually an algebraic expression involving variables and constants. To find the product of (8-6b)(5-3b), we need to use the distributive property of multiplication.
We can start by multiplying 8 by 5, which gives us 40. Next, we multiply 8 by -3b, which gives us -24b. Then, we multiply -6b by 5, which gives us -30b. Finally, we multiply -6b by -3b, which gives us[tex]18b^2[/tex].
Putting all of these terms together, we get:
(8-6b)(5-3b) = [tex]40 - 24b - 30b + 18b^2[/tex]
Simplifying this expression, we can combine the like terms -24b and -30b to get -54b. So the final answer is:
(8-6b)(5-3b) = [tex]18b^2 - 54b + 40[/tex]
Therefore, the product of (8-6b)(5-3b) is [tex]18b^2 - 54b + 40[/tex].
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In ΔSTU, t = 3. 4 cm, u = 6. 9 cm and ∠S=21°. Find the area of ΔSTU, to the nearest 10th of a square centimeter.
4.20 cm² is the area of the triangle STU to the nearest 10th of a square centimeter.
Given, t = 3.4 cm, u = 6.9 cm and ∠S = 21°
We know that the formula for the area of the triangle = 1/2 * t*u *sin(S)
Substituting the values
Area = 1/2 × 3.4 × 6.9 × sin(21°)
Area = 11.73 × sin(21°)
Area = 11.73 × 0.3583
Area = 4.2016
Rounding to the nearest 10th of a square centimeter .
Area = 4.20 cm²
Hence, 4.20 cm² is the area of the triangle STU to the nearest 10th of a square centimeter.
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Let the region r be the area enclosed by the function f(x)=x^2+1 and g(x)=2x+1. if the region r is the base of a solid such that each cross section perpendicular to the x axis is a square, find the volume of the solid.
The volume of the solid is 32/15 cubic units.
How to find volume of the solid?To find the volume of the solid, we need to integrate the area of each square cross section perpendicular to the x-axis over the interval [a, b], where a and b are the x-coordinates of the intersection points of f(x) and g(x):
First, we find the intersection points of the two functions:
x²+1 = 2x+1
x² - 2x = 0
x(x-2) = 0
x = 0 or x = 2
So, a = 0 and b = 2.
Next, we find the side length of each square cross section. Since the cross section is a square, the side length is equal to the difference between the y-coordinates of the functions f(x) and g(x) at each x:
Side length = f(x) - g(x) = (x²+1) - (2x+1) = x² - 2x
Finally, we integrate the area of each square cross section over the interval [0, 2] to get the volume of the solid:
V = ∫[0,2] (x² - 2x)² dx
V = ∫[0,2] (x⁴- 4x³ + 4x²) dx
V = [1/5 x⁵ - 1 x⁴ + 4/3 x³] [0,2]
V = (1/5 x⁵ - 1 x⁴ + 4/3 x³)|[0,2]
V = (1/5(2⁵) - 1(2⁴) + 4/3(2³)) - (1/5(0⁵) - 1(0⁴) + 4/3(0³))
V = (32/5 - 16/3)
V = 32/15
Therefore, the volume of the solid is 32/15 cubic units.
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2. Calculate the volume of the solid by calculating the triple integral: 6 pts •1 r2-2y dzdydx y = d x=0 +2=2 =3 y 3 =0
The volume of the solid is given by V = 9r/2 - 3.
To calculate the volume of this solid, we will use a triple integral, which involves integrating a function of three variables over a three-dimensional region. The triple integral is denoted by ∭f(x, y, z) dV, where f(x, y, z) is the function we are integrating, and dV is the volume element.
In our problem, the function f(x, y, z) is equal to 1, which means we are integrating a constant function. Therefore, we can simplify the triple integral to V = ∭dV, where V represents the volume of the solid.
To evaluate the triple integral, we need to determine the limits of integration for each variable. We are given the limits for x, y, and z, so we can set up the triple integral as follows:
V = ∫₂⁰ ∫₃⁰ r2-2y 1 dz dy dx
We integrate first with respect to z, then y, and finally x.
Integrating with respect to z, we get:
V = ∫₂⁰ ∫₃⁰[r2-2y - 1] dy dx
Simplifying the integral, we get:
V = ∫₂⁰ [r2y - y2]dy dx
Integrating with respect to y, we get:
V = ∫₂⁰ [(r2/2)y2 - (1/3)y3]dy
Simplifying the integral, we get:
V = [(r/2)(3)2 - (1/3)(3)3] - [(r2/2)(0)2 - (1/3)(0)3]
V = 9r/2 - 3
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Emmanuel weighs 150 pounds and he has normal liver function. It will take him approximately ________ hours to metabolize one standard drink
Emmanuel weighs 150 pounds and he has normal liver function. It will take him approximately 1 hour to metabolize one standard drink. Metabolism of alcohol is primarily done in the liver where it is broken down into acetaldehyde, which is then further broken down into water and carbon dioxide.
The liver can only metabolize a certain amount of alcohol per hour, which is why it takes time for the body to process and eliminate alcohol. However, other factors such as age, gender, body composition, and food consumption can also affect how quickly alcohol is metabolized.
It is important to drink responsibly and be aware of how alcohol can affect your body.
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Describe and correct the error a student made in finding the domain for the quotient when f(x) = 2x² - 3x + 1 and g(x) = 2x - 1.
So the domain is all real numbers.
y= 3x-2 y= 9x+ 10 find x, y
Answer:
(-2,-8)
Step-by-step explanation:
First, we have to make these linear equations into standard form:
-3x+y=-2
and
9x-y=-10
Now we tell my using elimination method, we can cross out the y variables because when added(y+(-y)) is just 0, so we just cross them out
Add liked terms
6x=-12
Solve for X:
X=-2
Plug 2 for X in any equation (lets do -3x+y=-2)
Plug in -2 for X:
-3(-2)+y=-2
Thus we get 6+y=-2
Solve for Y:
y=-8
Now that we have both our variables, we know that the answer is (-2,-8)
If x -1/x=3 find x cube -1/xcube
Answer:
Sure. Here are the steps on how to solve for x^3 - 1/x^3:
1. **Cube both sides of the equation x - 1/x = 3.** This will give us the equation x^3 - 3x + 1/x^3 = 27.
2. **Subtract 1 from both sides of the equation.** This will give us the equation x^3 - 1/x^3 = 26.
3. **The answer is 26.**
Here is the solution in detail:
1. **Cube both sides of the equation x - 1/x = 3.**
```
(x - 1/x)^3 = 3^3
```
```
x^3 - 3x + 1/x^3 = 27
```
2. **Subtract 1 from both sides of the equation.**
```
x^3 - 1/x^3 - 1 = 27 - 1
```
```
x^3 - 1/x^3 = 26
```
3. The answer is 26.
Question: Nora needs to cut some equal pieces of yarn for her Science project. The piece of yarn she has is 67. 6 inches long. Each piece of yarn must be 1. 3 inches in lenght. How many pieces of yarn will Nora have.
Nora will be able to cut 52 equal pieces of yarn for her Science project.
To find out how many equal pieces of yarn Nora can cut for her Science project, we need to divide the total length of the yarn by the length of each piece.
Total length of yarn: 67.6 inches
Length of each piece: 1.3 inches
Step 1: Divide the total length by the length of each piece.
67.6 inches ÷ 1.3 inches = 52
Nora will have 52 equal pieces of yarn, each 1.3 inches long, for her Science project.
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find cif a = 2.74 mi, b = 3.18 mi and ZC = 41.9°. Enter c rounded to 2 decimal places. C= mi Assume LA is opposite side a, ZB is opposite side b, and ZC is opposite side c.
Cif a = 2.74 mi, b = 3.18 mi and ZC = 41.9° and c^2 = a^2 + b^2 - 2ab*cos(C)where C is the angle opposite to side c, c comes to be ≈ 4.26 mi.
The Law of Cosines is a numerical formula that relates the side lengths and points of any triangle. It expresses that the square of any side of a triangle is equivalent to the number of squares of the other different sides short two times the result of those sides and the cosine of the point between them. To get side c, we can use the law of cosines, which states that c² = a² + b² - 2ab cos(C).
Plugging in the given values, we get:
c² = (2.74)² + (3.18)² - 2(2.74)(3.18)cos(41.9°)
c² ≈ 18.126
Taking the square root of both sides, we get:
c ≈ 4.26 mi
Rounding to 2 decimal places, c ≈ 4.26 mi.
Therefore, the answer is: c ≈ 4.26 mi.
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for each rectangle find the radio of the longer side to shorter side
[tex]\cfrac{\stackrel{\textit{longer side}}{12}}{\underset{\textit{shorter side}}{3\sqrt{3}}}\implies \cfrac{4}{\sqrt{3}}\implies \cfrac{4}{\sqrt{3}}\cdot \cfrac{\sqrt{3}}{\sqrt{3}}\implies \cfrac{4\sqrt{3}}{3}[/tex]
The spies of Syracuse report that enemies are marching towards the city. Archimedes needs to build death rays and claws to defend the city with. He'll need at least 10 machines but the city only gave him 3000 lbs of gold to build the machines with. A claw costs 200 lbs of gold to build while a death ray is worth 350 lbs of gold. Write a system of inequalities to find a possible number of claws and death rays that Archimedes can build. â
Possible number of death rays (D) and claws (C) that Archimedes can build are given by the following system of inequalities: 350D + 200C ≤ 3000. D, C ≥ 0
The first inequality represents the fact that the total amount of gold used to build the machines cannot exceed the 3000 lbs of gold given by the city. The second inequality ensures that the number of death rays and claws cannot be negative.
To explain this system, let us assume that Archimedes builds x death rays and y claws. The amount of gold required to build x death rays and y claws is given by 350x + 200y. The first inequality ensures that this value cannot exceed 3000 lbs of gold. The second inequality ensures that the number of death rays and claws cannot be negative.
Therefore, the solution to this system of inequalities gives us all the possible combinations of death rays and claws that Archimedes can build with the given amount of gold.
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Juanita is making a ribbon as shown 4 cm 15 cm 3 cm explain two different ways you can find the area of the ribbon then find the area of the ribbon
Answer:
6
Step-by-step explanation:
A=hbb/2=4·3/2=6
To find a triangle's area, use the formula area = 1/2 * base * height. Choose a side to use for the base, and find the height of the triangle from that base. Then, plug in the measurements you have for the base and height into the formula
or
The area of a triangle is the space enclosed within the three sides of a triangle. It is calculated with the help of various formulas depending on the type of triangle and is expressed in square units like, cm2, inches2, and so on.
HELPPPPPPPPPp WILL GIVE BRAINLEISTTT!!!
Answer:
100
Step-by-step explanation:
i think this is right
Can someone please answer numbers 12, 13, 14, and 15?
The average price of a two-bedroom apartment in the uptown area of a prominent American city during the real estate boom from 1994 to 2004 can be approximated by p(t) = 0.17e⁰.¹⁰ᵗ million dollars (0 ≤ t ≤ 10) where t is time in years (t = 0 represents 1994). What was the average price of a two-bedroom apartment in this uptown area in 2002, and how fast was it increasing? (Round your answers to two significant digits.) p(8) = $ million p' (8) = $ million per yr
In 2002, the average price of a two-bedroom apartment in the uptown area was approximately $0.316 million, and it was increasing at a rate of about $0.0328 million per year.
To find the average price of a two-bedroom apartment in 2002 (t=8), you need to evaluate the given function p(t) = 0.17e^(0.10t) at t=8:
p(8) = 0.17e^(0.10 * 8)
p(8) = 0.17e^0.8 ≈ 0.316 million dollars
To find the rate at which the price was increasing in 2002, you need to find the derivative of the function p(t) with respect to t, and then evaluate it at t=8:
p'(t) = d/dt (0.17e^(0.10t))
p'(t) = 0.17 * 0.10 * e^(0.10t)
Now, evaluate p'(t) at t=8:
p'(8) = 0.17 * 0.10 * e^(0.10 * 8)
p'(8) ≈ 0.0328 million dollars per year
So, in 2002, the average price of a two-bedroom apartment in the uptown area was approximately $0.316 million, and it was increasing at a rate of about $0.0328 million per year.
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Medical records at a doctor’s office reveal that 12% of adult patients have seasonal allergies. Select a random sample of 100 adult patients and let p^ = the proportion of individuals in the sample who have allergies.
(a) Calculate the mean and standard deviation of the sampling distribution of p^.
(b) Interpret the standard deviation from part (a).
(c) Would it be appropriate to use a normal distribution to model the sampling distribution of p^ ? Justify your answer
We know that both np and nq are greater than or equal to 10, it is appropriate to use a normal distribution to model the sampling distribution of p^ in this case.
(a) To calculate the mean and standard deviation of the sampling distribution of p^, we'll use the formulas:
Mean (μ) = p
Standard deviation (σ) = √(p*q/n)
where p is the proportion of individuals with allergies (0.12), q is the proportion of individuals without allergies (1 - p = 0.88), and n is the sample size (100).
Mean (μ) = 0.12
Standard deviation (σ) = √(0.12 × 0.88 / 100) = 0.02887
(b) The standard deviation of 0.02887 in this context represents the variability in the proportions of individuals with allergies that we would expect to observe in different random samples of 100 adult patients each. It quantifies the dispersion of p^ around the true population proportion.
(c) To determine if it's appropriate to use a normal distribution to model the sampling distribution of p^, we can check if both np and nq are greater than or equal to 10:
np = 100 × 0.12 = 12
nq = 100 × 0.88 = 88
Since both np and nq are greater than or equal to 10, it is appropriate to use a normal distribution to model the sampling distribution of p^ in this case.
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You have a machine which can paint 20 bikes per hour. you purchase two additional, identical machines. how many bikes can you now paint per hour
The total number of bikes that can be painted in an hour would be 60 bikes.
With three identical machines,
the number of bikes machine can paint per hour = 20,
the number of machines bought again = 2,
so the total number of machines will be = 3,
when there are two same machines the productivity will be = 20 * 3 = 60 bikes.
This is because each machine works independently and can paint bikes simultaneously.
By adding two additional machines to the existing one,
the productivity of the painting process can be significantly increased. The new machines will not only increase the overall capacity but also reduce the turnaround time required for painting a large number of bikes.
By investing in additional machines,
the business can increase its output and generate more revenue,
which can be used to expand the operations further.
It's important to note that the investment in additional machines needs to be justified by the demand for painted bikes and the expected return on investment.
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please show all steps :)
For the following system: Determine how, if at all, the planes intersect. If they do, determine the intersection. [2T/3A] 2x + 2y + z - 10 = 0 5x + 4y - 4z = 13 3x – 2z + 5y - 6 = 0
The planes intersect at the point (-19/21, -11/14, 1).
How to find intersection of three planes in three-dimensional space?To determine how, if at all, the planes intersect, we need to solve the system of equations given by the three planes:
[2T/3A] 2x + 2y + z - 10 = 0
5x + 4y - 4z = 13
3x – 2z + 5y - 6 = 0
We can use elimination to solve this system. First, we can eliminate z from the second and third equations by multiplying the second equation by 2 and adding it to the third equation:
5x + 4y - 4z = 13
6x - 4z + 10y - 12 = 0
11x + 14y - 12 = 0
Next, we can eliminate z from the first and second equations by multiplying the first equation by 2 and subtracting the second equation from it:
4x + 4y + 2z - 20 = 0
-5x - 4y + 4z = -13
9x - y - 6z - 20 = 0
Now we have two equations in three variables. To eliminate y, we can multiply the second equation by 14 and subtract it from the first equation:
11x + 14y - 12 = 0
-70x - 56y + 56z = -182
-59x - 42z - 12 = 0
Finally, we can substitute this expression for x into one of the previous equations to find z:
3(59/42)z - 12/42 - 2y - 10 = 0
177z - 60 - 84y - 420 = 0
177z - 84y - 480 = 0
Now we have two equations in two variables, z and y. We can solve for y in terms of z from the second equation:
y = (177/84)z - (480/84)
Substituting this expression for y into the third equation, we can solve for z:
177z - 84[(177/84)z - (480/84)] - 480 = 0
177z - 177z + 480 - 480 = 0
This equation simplifies to 0=0, which means that z can be any value. Substituting z=1 into the expression for y, we get:
y = (177/84)(1) - (480/84) = -11/14
Substituting z=1 and y=-11/14 into the expression for x, we get:
x = (59/42)(1) - (12/42) + 2(-11/14) + 10 = -19/21
Therefore, the planes intersect at the point (-19/21, -11/14, 1).
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The area of the shaded region under the curve of a function f(x) = ax + b on the interval [ 0, 4 ] is 16 square units.
The area of the given region under the curve of a function f(x) = ax + b on the interval [0, 4] is 16 square units. So, all the options satisfy the value of a and b except (7, -9).
How to find the area of a region?The area of the given region under the curve of a function f(x) = ax + b on the interval [0, 4] is 16 square units.
f(x) is a linear function, the area under the curve on the interval [0,4] is a trapezoid with a height of 4 and bases of lengths f(0) and f(4).
The area of a trapezoid is the height times the average of the bases.
a. f(x)=-2x+8 f(0)=8, f(4)=0;
area = 4(8/2) = 16
b. f(x)=x+2; f(0)=2, f(4)=6;
area = 4(8/2) = 16
c. f(x)=3x-2; f(0)=-2, f(4)=10;
area = 4(8/2) = 16
d. f(x)=5x-6; f(0)=-6, f(4)=14;
area = 4(8/2) = 16
e. f(x)=7x-9; f(0)=-9, f(4)=19;
area = 4(10/2) = 20
Thus, The area is NOT 16 for choice e.
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the length of a retangle is 5 more than twoce its width its perimter is 88 feet find the dimensions use p=2l+2w
If length of rectangle is 5 more than twice it's width and having perimeter as 88 feet, then the dimensions of the rectangle are, length is 31 feet, width is 13 feet.
Let width of the rectangle be represented as = "w" feet.
It is given that, the length of rectangle is 5 more than twice it's width,
So, Length can be represented as "2w + 5" in feet;
We use formula for perimeter of rectangle, which is "P = 2Length + 2Width", where P = perimeter, L = length, and W = width.
In this case, we know that the perimeter is 88 feet, so we substitute the values,
We get,
⇒ 88 = 2(2w + 5) + 2w;
⇒ 88 = 4w + 10 + 2w,
⇒ 88 = 6w + 10,
⇒ 78 = 6w,
⇒ w = 13,
So the width is 13 feet. We use this value of width to find length of the rectangle:
⇒ L = 2w + 5,
⇒ L = 2(13) + 5,
⇒ L = 31
Therefore, the dimensions of the rectangle are 31 feet by 13 feet.
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Evan bought 7 books on sale for 45.50 the regular price of the 7 books 57.75 how much did evan save per books buying them on salw
Evan saved $1.75 per book by buying them on sale.
Evan bought 7 books on sale for $45.50, with a regular price of $57.75. What was the per-book savings?To find out how much Evan saved per book by buying them on sale, you can use the following formula:
Savings per book = (Regular price per book - Sale price per book)
First, you need to find the regular price per book:
Regular price per book = (Total regular price of 7 books) / 7
Regular price per book = 57.75 / 7
Regular price per book = 8.25
Next, you need to find the sale price per book:
Sale price per book = (Total sale price of 7 books) / 7
Sale price per book = 45.50 / 7
Sale price per book = 6.50
Now, you can find the savings per book:
Savings per book = (Regular price per book - Sale price per book)
Savings per book = (8.25 - 6.50)
Savings per book = 1.75
Therefore, Evan saved $1.75 per book by buying them on sale.
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