Answer:
Step-by-step explanation:
The translation theorem of transforms states that if F(s) has a Laplace transform, then the Laplace transform of e^at F(s) is given by F(s-a) for s > a.
Using this theorem, we can find the inverse Laplace transform of 9s + 10 F(s) by first writing it in the form e^at F(s-a), where a is the constant term in the expression 9s + 10 F(s).
Since the Laplace transform of a constant is given by L{1} = 1/s, we can write:
9s + 10 F(s) = 9s + 10 L{1/s} F(s)
= 9s + 10 ∫ e^(-st)/t dt F(s)
= 9s + 10 ln(s) F(s)
Comparing this to the form e^at F(s), we see that a = ln(s), so we can write:
9s + 10 F(s) = e^ln(s) F(s) = s F(s-ln(s))
Thus, applying the translation theorem, we have:
L^-1{9s + 10 F(s)} = L^-1{s F(s-ln(s))} = e^t f(t-ln(t))
where f(t) is the inverse Laplace transform of F(s).
In summary, we can find the inverse Laplace transform of 9s + 10 F(s) by first expressing it in the form e^at F(s), finding the value of a, and then applying the translation theorem to obtain e^t f(t-ln(t)).
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the inverse Laplace transform
9 δ'(t) + 65 [tex]e^{-58t}[/tex]- 65 δ(t)
What is the translation theorem?According to the translation theorem, if the Laplace transform of a function f(t) is F(s), then the Laplace transform of the function [tex]e^{-at}[/tex]f(t) is given by F(s+a).
In this case, let's consider the function 9s + 10 F(s) - 65/(s+58). We can rewrite this as:
9s + 10 F(s) - 65/(s+58) = 9s + 10 [F(s) - 6.5/(s+58)]
Notice that we have a term in square brackets that looks like the Laplace transform of a function [tex]e^{-at}[/tex]f(t) with a=58 and f(t)=6.5. As a result, the translation theorem can be used to determine this term's inverse Laplace transform, which we can then combine with the 9s inverse Laplace transform to obtain the final result.
So, we have:
[tex]L^{-1}{10[F(s) - 6.5/(s+58)]}[/tex] = 10 [tex]L^{-1}{F(s) - 6.5/(s+58)}(t)[/tex]
= 10 [f(t) - 6.5 e^(-58t)] = 10 [6.5 δ(t) - [tex]6.5 e^{-58t}[/tex])]
where δ(t) is the Dirac delta function.
We can write the inverse Laplace transform of the original function as 9δ'(t), where δ'(t) is the derivative of the Dirac delta function, using the inverse Laplace transform of 9s:
L^-1{9s + 10 F(s) - 65/(s+58)} = 9 δ'(t) + 65 [tex]e^{-58t}[/tex]- 65 δ(t)
Therefore, the inverse Laplace transform of 9s + 10 F(s) - 65/(s+58) is given by:
9 δ'(t) + 65 [tex]e^{-58t}[/tex]- 65 δ(t)
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To warm up, Coach Hadley had his swim team swim twelve 5–meter long laps. It took the team 5 minutes to finish the warm up. How fast did the team swim in centimeters per second?
The team swam at a speed of 20 centimeters per second during their warm up.
First, let's convert the length of one lap from meters to centimeters:
5 meters = 500 centimeters
So, the team swam 12 laps of 500 centimeters each, for a total distance of:
12 laps × 500 centimeters/lap = 6000 centimeters
Next, let's convert the time from minutes to seconds:
5 minutes = 300 seconds
To find the speed in centimeters per second, we can divide the distance by the time:
speed = distance ÷ time = 6000 centimeters ÷ 300 seconds
simplifying, we get:
speed = 20 centimeters/second
Therefore, the team swam at a speed of 20 centimeters per second during their warm up.
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Will give a lot of points
Arturo launches a toy rocket from a platform. The height of the rocket in feet is given
by h(t)=-16t² + 32t + 128 where t represents the time in seconds after launch.
What is the rocket's greatest height?
Arturo launches a toy rocket from a platform. The height of the rocket in feet is given, so the rocket's greatest height is 144 feet.
The greatest height of the rocket corresponds to the vertex of the parabolic function h(t) = -16t² + 32t + 128, which occurs at the time t = -b/(2a), where a = -16 and b = 32.
So, t = -b/(2a) = -32/(2*(-16)) = 1.
Therefore, the rocket's greatest height occurs after 1 second of launch. We can find the height by substituting t = 1 into the equation for h(t):
h(1) = -16(1)² + 32(1) + 128 = 144.
Therefore, the rocket's greatest height is 144 feet.
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Match the word(s) with the descriptive phrase.
1. a polyhedron with two congruent faces that lie in parallel planes
2. the sum of the areas of the faces of a polyhedron
3. the faces of a prism that are not bases
4. the sum of the areas of the lateral faces
5. a solid with two congruent circular bases that lie in parallel planes
A. lateral area
. B. lateral faces
C. prism
D. surface area
E. cylinder
Answer:
Step-by-step explanation:
1. B. lateral faces
2. D. surface area
3. B. lateral faces
4. A. lateral area
5. E. cylinder
How many lines can be
constructed through point P
that are perpendicular to AB?
Answer:
A. 2
Step-by-step explanation:
It would be a triangle. There's no other way unless you used a point in between a and b
The figure shows the dimensions of the cube-shaped box Amy uses to hold her
rings. What is the surface area of Amy's box?
A 1030.3 square centimeters
B 612.06 square centimeters
C 600 square centimeters
D 408.04 square centimeters
The surface area of Amy's box which is cube has is 294 square cm.
The surface area of a cube is given by the formula:
SA = 6s²
where s is the length of a side of the cube.
From the given figure, we can see that the length of a side of the cube is 7 cm.
Substituting s = 7 into the formula, we get:
SA = 6(7²)
= 294 square cm
Therefore, the surface area of Amy's box is 294 square cm.
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The side length of cube-shaped box is 7 cm find the surface area?
You make a pudding for a dinner party and put it in the refrigerator at 5 P.M. (t — 0). Your refrigerator maintains a constant temperature Of 400. The pudding will be ready to serve when it cools to 450. When you put the pudding in the refrigerator you measure its temperature to be 1900, and when the first guest arrives at 6 P.M., you measure it again and get a temperature reading of 1000. Based on Newton's Law of Cooling, when is the earliest you can serve the pudding?
The earliest time you can serve the pudding is t = (-ln(30)) / k.
Where k is the constant value.
We have,
To find the earliest time you can serve the pudding, we need to determine the time at which the temperature of the pudding reaches 450.
Using Newton's Law of Cooling, the equation for the temperature of the pudding at time t is given by:
[tex]T(t) = T_{ambient} + (T_{initial} - T_{ambient} \times e^{-kt}[/tex]
Where:
T(t) is the temperature of the pudding at time t,
T_ambient is the ambient temperature (400),
T_initial is the initial temperature of the pudding (1900),
k is the cooling constant,
t is the time.
To find the earliest time, we set T(t) equal to 450 and solve for t:
[tex]450 = 400 + (1900 - 400) \times e^{-kt}[/tex]
Simplifying the equation, we get:
[tex]e^{-kt} = (450 - 400) / (1900 - 400)\\e^{-kt} = 50 / 1500\\e^{-kt} = 1 / 30[/tex]
Taking the natural logarithm of both sides:
-ln(30) = -kt
Solving for t, we have:
t = (-ln(30)) / k
Without the specific value of the cooling constant k, we cannot determine the exact value of the earliest time to serve the pudding.
Thus,
The earliest time you can serve the pudding is t = (-ln(30)) / k.
Where k is the constant value.
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pls help me i’m struggling!!
Yolanda wants to replace the grass in this triangular section of her yard with mulch. A bag of mulch costs $4.85 and covers 3 square feet. Which of the following statements accurately describe this situation? Select all that apply.
Yolanda wants to replace the grass in this triangular section of her yard with mulch. A bag of mulch costs $4.85 and covers 3 square feet. Which of the following statements accurately describe this situation? Select all that apply.
Answer:
Step-by-step explanation:
Chelsea has 2. 24 pounds of meat. She uses 0. 16 pound of meat to make one hamburger. How many hamburgers can Chelsea make with the meat she has?
Answer:
14 hamburgers
Step-by-step explanation:
The problem uses division, but we can create a proportion to see how the division works.
Since we know that Chelsea can make 1 hamburger with 0.16 pounds and allow x to represent the number of burgers Chelsea can make with 2.24 lbs of meat, we have:
[tex]\frac{2.24}{x}=\frac{0.16}{1}[/tex]
[tex]2.24=0.16x[/tex]
As the proportion shows, we can divide 2.24 by 0.16 to get x = 14.
Check: 0.16 lbs * 14 patties = 2.24 lbs
consider all 5 letter "words" made from the letters a through h. (recall, words are just strings of letters,not necessarily actual english words.)(a) how many of these words are there total?
There are a total of 32768 (8) 5-letter "words" made from the letters a through h when all possible combinations are considered.
Consider all 5-letter "words" made from the letters A through H. There are a total of 8 unique letters, and since repetition is allowed, you can form 8 different possibilities for each of the 5 positions in the word. To calculate the total number of these words, simply multiply the possibilities for each position: 8 * 8 * 8 * 8 * 8 = 32,768. So, there are 32,768 possible 5-letter "words" using the letters A through H.
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Matt bought a collection of 1660 stamps. He needs to choose between an album with large pages and an album with small pages to hold his stamps. The number of stamps per page for both album sizes is shown in the table. How many of each type of page will Matt need to hold all 1660 stamps?
Answer:Martin has 2212 stamps.
Step-by-step explanation:
Given,
Total number of pages = 48,
Out of which first 20 pages each have 35 stamps in 5 rows,
So, the stamps in first 20 pages = 35 × 20 = 700,
Now, the remaining number of pages = 48 - 20 = 28,
Also, each remaining page has 54 stamps,
So, the total stamps contained by remaining pages = 54 × 28 = 1512,
Hence, total stamps = stamps in 20 pages + stamps in 28 pages
= 700 + 1512
= 2212
Step-by-step explanation:
Edwin's soccer team has a tradition of going out for pizza after each game. Last week, the team ordered 2 medium pizzas and 4 large pizzas for a total of 56 slices. This week, the team ordered 3 medium pizzas and 3 large pizzas for a total of 54 slices.
Session 3
(Calculator)
David just bought six more baseball cards. The new baseball cards represent 30% of
David's special edition baseball card collection.)
Number of Baseball Cards
6
++
0
+
25 30
+
50
+
75
?
+
100
What is the total number of cards in David's baseball card collection?
Enter your answer in the box.
Answer:
If the new baseball cards represent 30% of David's special edition baseball card collection, then the original collection represents 70%. Let's represent the total number of cards in David's collection with the variable x. Then we can set up the following equation:
6 = 0.3x
To solve for x, we can divide both sides by 0.3:
x = 6 ÷ 0.3 = 20
Therefore, the total number of cards in David's baseball card collection is 20.
if a is a square matrix there exists a matrix b such that ab equals the identity matrix. T/F
True. This is a true statement known as the invertible matrix theorem. If a square matrix is invertible, then there exists a matrix b such that ab equals the identity matrix. However, not all square matrices are invertible.
True. If matrix A is a square matrix and has an inverse matrix B, then the product of A and B (AB) equals the identity matrix. In other words, if A is invertible, there exists a matrix B such that AB = BA = I, where I is the identity matrix. This is a true statement known as the invertible matrix theorem. If a square matrix is invertible, then there exists a matrix b such that ab equals the identity matrix. However, not all square matrices are invertible.
True. If matrix A is a square matrix and has an inverse matrix B, then the product of A and B (AB) equals the identity matrix. In other words, if A is invertible, there exists a matrix B such that AB = BA = I, where I is the identity matrix.
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determine the identity to (1 - (sin(x) - cos(x))^2)/(2 cos(x))a. tan (x) b. cos (x)c. sec (a)d. sin(x) e. none of these
The identity is (d) sin(x).
We can start by expanding the numerator:
(1 - (sin(x) - cos(x))^2) = 1 - (sin^2(x) - 2sin(x)cos(x) + cos^2(x))
= 1 - (1 - sin(2x))
= sin(2x)
Therefore, the expression simplifies to:
sin(2x)/(2cos(x))
Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x), we get:
2sin(x)cos(x)/(2cos(x)) = sin(x)
So the identity is (d) sin(x).
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for what values of b are the given vectors orthogonal? (enter your answers as a comma-separated list.) −11, b, 2 , b, b2, b
The given vectors are:
Vector A = (-11, b, 2)
Vector B = (b, b^2, b)
The values of b for which the given vectors are orthogonal are 0, -3, and 3.
Dot product:
To find the values of b for which the given vectors are orthogonal, we need to use the dot product of the vectors.
To determine if two vectors are orthogonal, their dot product should be equal to zero.
The dot product is calculated as follows:
Dot Product (A, B) = A1 * B1 + A2 * B2 + A3 * B3
Substituting the components of Vector A and Vector B:
(-11 * b) + (b * b^2) + (2 * b) = 0
Now, simplify the equation:
-11b + b^3 + 2b = 0
b^3 - 9b = 0
Factor the equation:
b(b^2 - 9) = 0
Now, we can find the values of b:
b = 0
b^2 - 9 = 0
b^2 = 9
b = ±3
So, the values of b for which the given vectors are orthogonal are 0, -3, and 3.
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an observer views the space shuttle from a distance of x = 2 mi from the launch pad.(a) Express the height of the space shuttle as a function of the angle of elevation θ. (b) Express the angle of elevation as a function of the height h of the space shuttle.
The angle of elevation is a function of the height of the space shuttle given by θ = arctan(h / 2).
Angle of elevation calculation.
(a) To express the height of the space shuttle as a function of the angle of elevation θ, we can use trigonometry. Let h be the height of the space shuttle above the launch pad. Then, we have:
tan(θ) = h / x
Solving for h, we get:
h = x * tan(θ)
Substituting x = 2 mi, we get:
h = 2 * tan(θ) mi
Therefore, the height of the space shuttle is a function of the angle of elevation θ given by h = 2 * tan(θ) mi.
(b) To express the angle of elevation as a function of the height h of the space shuttle, we rearrange the equation we found in part (a) as follows:
tan(θ) = h / x
tan(θ) = h / 2
Taking the inverse tangent of both sides, we get:
θ = arctan(h / 2)
Therefore, the angle of elevation is a function of the height of the space shuttle given by θ = arctan(h / 2).
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how many arrangements of mathematics are there in which each consonant is adjacent to a vowel?
There are 1,152 arrangements of "MATHEMATICS" in which each consonant is adjacent to a vowel.
To find the number of arrangements of the word "MATHEMATICS" in which each consonant is adjacent to a vowel, we can treat the consonants (M, T, H, M, T, C, and S) and vowels (A, E, A, I) as separate groups and arrange them in a way that each group of consonants is next to a group of vowels
We have two groups of vowels (A and E, and A and I) and three groups of consonants (MTHM, TC, and S). We can arrange the two vowel groups in 2! = 2 ways, and then arrange the three consonant groups in 3! = 6 ways. Within each group, the letters can be arranged in a total of 4! = 24 ways for the MTHM group, 2! = 2 ways for the TC group, and 1 way for the S group.
Therefore, the total number of arrangements in which each consonant is adjacent to a vowel is:
2! x 6 x 24 x 2 x 1 = 1,152
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a random variable x is normally distributed with µ = 185 and σ = 14. find the 72th percentile of the distribution. round your answer to the tenths place.
The 72nd percentile of the normally distributed random variable x is 196.788, rounded to the tenths place.
What is percentile?Percentile is expressed as a percentage of the group that is equal to or lower than the individual in question.
The 72nd percentile of a normally distributed random variable x is calculated by using the formula z = (x - μ) / σ, where z is the z-score, μ is the mean of the data, and σ is the standard deviation of the data.
In this case, z = (x - 185) / 14.
To find the 72nd percentile, we have to use the z-score table to look up the z-score that corresponds to the desired percentile.
The z-score for the 72nd percentile is 0.842.
Plugging this back into our formula, we get
x = 185 + (0.842 * 14)
= 196.788.
Therefore, the 72nd percentile of the normally distributed random variable x is 196.788, rounded to the tenths place.
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The 72nd percentile of the normally distributed random variable x is 196.788, rounded to the tenths place.
What is percentile?Percentile is expressed as a percentage of the group that is equal to or lower than the individual in question.
The 72nd percentile of a normally distributed random variable x is calculated by using the formula z = (x - μ) / σ, where z is the z-score, μ is the mean of the data, and σ is the standard deviation of the data.
In this case, z = (x - 185) / 14.
To find the 72nd percentile, we have to use the z-score table to look up the z-score that corresponds to the desired percentile.
The z-score for the 72nd percentile is 0.842.
Plugging this back into our formula, we get
x = 185 + (0.842 * 14)
= 196.788.
Therefore, the 72nd percentile of the normally distributed random variable x is 196.788, rounded to the tenths place.
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find the inflection point of the function. (hint: g ' ' ( 0 ) g′′(0) does not exist.) g ( x ) = 9 x | x |
The inflection point of g(x) = 9|x| is x = 0.
First, let's find the first derivative of g(x):
g'(x) = 9 |x| + 9x * d/dx(|x|)
g'(x) = 9 |x| + 9x * sign(x)
where sign(x) is the sign function, which equals -1 for x < 0, 0 for x = 0, and 1 for x > 0.
Now, let's find the second derivative of g(x):
g''(x) = 9 * d/dx(|x|) + 9 * sign(x) + 9x * d/dx(sign(x))
g''(x) = 0 + 9 * sign(x) + 9x * d/dx(sign(x))
The derivative of the sign function is not defined at x = 0, but we can use the definition of the derivative to find the left and right limits of g''(x) as x approaches 0:
g''(0-) = lim x→0- [g''(x)]
g''(0-) = lim x→0- [9 * (-1) + 9x * (-∞)]
g''(0-) = -∞
g''(0+) = lim x→0+ [g''(x)]
g''(0+) = lim x→0+ [9 * (1) + 9x * (∞)]
g''(0+) = ∞
Since the left and right limits of g''(x) as x approaches 0 are not equal, g''(0) does not exist, and g(x) does not have an inflection point. Instead, the function changes concavity at x = 0, which is a vertical point of inflection.
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help me by answering this math question!!! i’ll mark brainliest
Answer:
30
because 1=3 so each of them are 30
Use logarithmic differentiation to find the derivative of the function. y = (x^3 + 2)^2(x^4 + 4)^4
The derivative of the function y = (x^3 + 2)^2(x^4 + 4)^4 using logarithmic differentiation is: y' = 2(x^3 + 2)(x^4 + 4)^3[3x^2(x^4 + 4) + 8x(x^3 + 2)^2]
To use logarithmic differentiation, we take the natural logarithm of both sides of the equation and then differentiate with respect to x using the rules of logarithmic differentiation.
ln(y) = ln[(x^3 + 2)^2(x^4 + 4)^4]
Now, we use the product rule and chain rule to differentiate ln(y):
d/dx [ln(y)] = d/dx [2ln(x^3 + 2) + 4ln(x^4 + 4)]
Using the chain rule, we get:
d/dx [ln(y)] = 2(1/(x^3 + 2))(3x^2) + 4(1/(x^4 + 4))(4x^3)
Simplifying this expression, we get:
d/dx [ln(y)] = 6x^2/(x^3 + 2) + 16x^3/(x^4 + 4)
Finally, we use the fact that d/dx [ln(y)] = y'/y to solve for y':
y' = y(d/dx [ln(y)])
Substituting in the expression for d/dx [ln(y)], we get:
y' = (x^3 + 2)^2(x^4 + 4)^4 [6x^2/(x^3 + 2) + 16x^3/(x^4 + 4)]
Simplifying this expression, we get:
y' = 2(x^3 + 2)(x^4 + 4)^3[3x^2(x^4 + 4) + 8x(x^3 + 2)^2]
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for each positive integer n, let p(n) be the formula 12 22 ⋯ n2=n(n 1)(2n 1)6. write p(1). is p(1) true?
The formula for p(n) is not valid for n = 1.
How to find p(1) is true?For each positive integer n, using the formula given, we can find p(1) by plugging in n = 1:
p(1) = 1(1-1)(2(1)-1)/6 = 0/6 = 0
So, according to the formula, p(1) is equal to 0.
However, we can see that this is not a true statement.
Because the product in the formula is defined as the product of the squares of the odd integers from 1 to n, and when n = 1, there is only one odd integer, which is 1.
Thus, p(1) should be equal to [tex]1^2 = 1.[/tex]
Therefore, the formula for p(n) is not valid for n = 1.
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If the shaded region is 1/6 of the perimeter of the circle with 10cm of the radius then find the measure of the angle inscribed in the circle.
The measure of the inscribed angle is determined as 150⁰.
What is the perimeter of the circle?
The perimeter of the circle is calculated as follows;
P = 2πr
where;
r is the radius of the circleP = 2π x 10 cm
P = 62.832 cm
The length of the shaded regions calculated as follows;
S = 1/6 x 62.832
S = 10.47 cm
The angle inscribed is calculated as follows;
θ/360 x 2πr = 10.47
2πrθ = 360 x 10.47
θ = ( 360 x 10.47 )/(2π x 10)
θ = 60⁰
angle at center = 360 - 60 = 300
inscribed angle = ¹/₂ x 300 (angle at center is twice angle at circumference)
inscribed angle = 150⁰
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a tree grows in height by 21% per
year. it is 2m tall after one year.
After how many more years will the
tree be over 20m tall
Answer:
12.08 years
Step-by-step explanation:
to overcome this problem we will have to use the exponential growth formula A = P(1+r)^t
where a is the final amount
where p is the initial amount
where r is the rate per year
where t is the number of years
we can say
20= 2(1+0.21)^t
solve the equation for t
20/2 = 2(1.21)^t/2
10 = 1.21^t
take the log of both sides we get that
t = 12.08 years
what is the volume in cubic centimeters of a right rectangular prism with a length of 10 cm, a width of 8 cm, and a height of 6 cm
Answer:
The formula for the volume of a right rectangular prism is given by:
Volume = Length * Width * Height
Given that the length is 10 cm, the width is 8 cm, and the height is 6 cm, we can substitute these values into the formula:
Volume = 10 cm * 8 cm * 6 cm
Multiplying the values:
Volume = 480 cubic centimeters
So, the volume of the right rectangular prism is 480 cubic centimeters.
According to the question, we were asked What is the volume, in cubic centimeters, of a right rectangular prism that has a length of 10 centimeters, a width of 8 centimeters, and a height of 6 centimeters?
When you hear about a rectangular prism, just know that we are talking about a cuboid and we all know here that the volume of a cuboid is the same as the volume of a rectangular prism which is:
[tex]\text{Length} \times \text{width} \times \text{breadth}[/tex].
And in this case, we have the length as 10 cm, the width as 8 cm and the subsequent height of the prism as 6 cm.
Applying this variables into the given formula for obtaining the volume for a prism,
We have [tex]9\times6\times10 = 480 \ \text{cm}^3[/tex]
Therefore, the volume of the right rectangular prism is 480 cm³.
The figure below shows a rectangle prism. One base of the prism is shaded
1. The volume of the prism is 144 cubic units
2. The area of the shaded base is 16units²
What is a prism?A prism is a solid shape that is bound on all its sides by plane faces. A prism can have a rectangular base( rectangular prism) or a triangular base( triangular prism) or a circular base ( cylinder) e.t.c
Generally the volume of a prism is expressed as;
V = base area × height.
base area = l × w
therefore volume = l× w ×h
The base area = l× w
= 8× 2 = 16 square units
therefore the volume of the prism = 16 × 9
= 144 cubic units
Therefore the volume of the prism is 144 cubic units and the shaded base area is 16 units².
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Help Evaluate function expressions
Thus,the solution of the expression for the given function f(x) and g(x) is found as: -1.f(-8) - 4.g(4) = -9.
Explain about the function:A function connects an element x to with an element f(x) in another set, to use the language of set theory. The domain and range of the function are the set of values of f(x) that are produced by the values there in domain of the function, which is the set of values of x.
This indicates that a function f will map an object x to just one object f(x) in a set of potential outputs if the object x is in the set of inputs known as the domain (called the codomain).The symbolism of a function machine, which accepts an object as its input and produces another entity as its output based on that input, makes the concept of a function simple to understand.given expression:
-1.f(-8) - 4.g(4) = ?
Get the value of f(-8) and g(4) from the graph shown.
f(-8) = -4
g(4) = 3
Put the expression:
= -1.*(-4) - 4*3
= 4 - 13
= -9
Thus,the solution of the expression for the given function f(x) and g(x) is found as: -1.f(-8) - 4.g(4) = -9.
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A couple of two-way radios were purchased from different stores. Two-way radio A can reach 6 miles in any direction. Two-way radio B can reach 12.88 kilometers in any direction.
Part A: How many square miles does two-way radio A cover? Use 3.14 for π and round to the nearest whole number. Show every step of your work. (3 points)
Part B: How many square kilometers does two-way radio B cover? Use 3.14 for π and round to the nearest whole number. Show every step of your work. (3 points)
Part C: If 1 mile = 1.61 kilometers, which two-way radio covers the larger area? Show every step of your work. (3 points)
Part D: Using the radius of each circle, determine the scale factor relationship between the radio coverages. (3 points)
A. Two-way radio A covers 113 square miles.
B. Rounded to the nearest whole number, two-way radio B covers 523 square kilometers.
C. Comparing the areas, we can see that radio B covers the larger area with 523 square kilometers.
D. The coverage area of radio B is approximately 1.33 times larger than the coverage area of radio A.
What is radius?Radius is a term used in geometry to describe the distance from the center of a circle or sphere to any point on its circumference or surface, respectively. It is usually denoted by the letter "r" and is measured in units of length, such as inches, centimeters, or meters. The radius of a circle is half of its diameter, while the radius of a sphere is one-half of its diameter.
Part A:
The area covered by two-way radio A can be calculated using the formula for the area of a circle:
Area = π x radius²
Radius of radio A = 6 miles
Area = 3.14 x 6²
Area = 113.04 square miles
Rounded to the nearest whole number, two-way radio A covers 113 square miles.
Part B:
The area covered by two-way radio B can also be calculated using the same formula:
Area = π x radius²
Radius of radio B = 12.88 kilometers
Area = 3.14 x (12.88)²
Area = 523.14 square kilometers
Rounded to the nearest whole number, two-way radio B covers 523 square kilometers.
Part C:
To compare the areas covered by the two-way radios, we need to convert the area covered by radio A from square miles to square kilometers, using the conversion factor given:
1 mile = 1.61 kilometers
Therefore, 1 square mile = (1.61)² square kilometers
Area covered by radio A = 113 square miles
Area covered by radio A in square kilometers = 113 x (1.61)²
Area covered by radio A in square kilometers = 290.22 square kilometers
Comparing the areas, we can see that radio B covers the larger area with 523 square kilometers.
Part D:
To determine the scale factor relationship between the radio coverages, we can divide the radius of radio B by the radius of radio A:
Scale factor = radius of radio B / radius of radio A
Scale factor = 12.88 kilometers / 6 miles
Scale factor = 12.88 kilometers / 9.66 kilometers (since 1 mile = 1.61 kilometers)
Scale factor = 1.33
This means that the coverage area of radio B is approximately 1.33 times larger than the coverage area of radio A.
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evaluate the line integral, where c is the given curve. c xey dx, c is the arc of the curve x = ey from (1, 0) to (e9, 9)
The line integral ∫C xey dx on the arc of the curve x = ey from (1, 0) to (e^9, 9) is (1/3)(e^27 - 1).
How to evaluate the line integral on the given curve?Hi! I'd be happy to help you evaluate the line integral on the given curve. To evaluate the line integral ∫C xey dx, where C is the arc of the curve x = ey from (1, 0) to (e^9, 9), follow these steps:
1. Parameterize the curve: Since x = ey, let y = t, so x = e^t. Thus, the parameterization of the curve is r(t) = (e^t, t), with t ranging from 0 to 9.
2. Compute the derivative of the parameterization: dr/dt = (de^t/dt, dt/dt) = (e^t, 1).
3. Substitute the parameterization into the integrand: xey = (e^t)(e^t) = e^(2t).
4. Compute the dot product of the integrand and dr/dt: (e^(2t)) * (e^t, 1) = e^(3t).
5. Integrate the dot product with respect to t from 0 to 9: ∫(e^(3t)) dt from t = 0 to t = 9.
6. Evaluate the integral: [1/3 * e^(3t)] from t = 0 to t = 9 = [1/3 * e^(27)] - [1/3 * e^0] = (1/3)(e^27 - 1).
So, the line integral ∫C xey dx on the arc of the curve x = ey from (1, 0) to (e^9, 9) is (1/3)(e^27 - 1).
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