If tan t =12/5 , and 0 ≤ t < π /2 , find sin t, cos t, sec t, csc t, and cot t.

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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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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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:

The formula for p(n) is not valid for n = 1.

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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Answer:

Step-by-step explanation:

1. B. lateral faces

2. D. surface area

3. B. lateral faces

4. A. lateral area

5. E. cylinder

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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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.

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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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.

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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The adjoint of the linear operator T on C^2 is T*(x,y) = (2x-2y, ix), and T*(3i,1+2i) = (4i-2, -3).

To compute T*(3 i, 1+2i), we need to first find the adjoint of T, denoted by T*. Recall that T* is the linear operator on C^2 such that for any (x,y) in C^2 and (a,b) in C^2, we have

<T*(x,y),(a,b)> = <(x,y),T(a,b)>

where <,> denotes the inner product on C^2.

To find T*, we need to compute <T*(x,y),(a,b)> for any (x,y) and (a,b) in C^2. We have

<T*(x,y),(a,b)> = <(x,y),T(a,b)>

= <(x,y),(2a+ib,(1-2)a)>

= 2ax + ibx + (1-2)ay

= (2a-2y)x + ibx

Thus, we see that T*(x,y) = (2x-2y, ix) for any (x,y) in C^2.

Now, to compute T*(3i,1+2i), we have

T*(3i,1+2i) = (2(3i)-2(1+2i), i(3i))

= (6i-2-4i, -3)

= (4i-2, -3)

Therefore, T*(3i,1+2i) = (4i-2, -3).

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The given question is incomplete, the complete question is:

Consider the linear operator T on C^2 defined by T(a, b) = (2a + ib, (1-2)a) Compute T*(3 i,1 2i)

The adjoint of the linear operator T on C^2 is T*(x,y) = (2x-2y, ix), and T*(3i,1+2i) = (4i-2, -3).

To compute T*(3 i, 1+2i), we need to first find the adjoint of T, denoted by T*. Recall that T* is the linear operator on C^2 such that for any (x,y) in C^2 and (a,b) in C^2, we have

<T*(x,y),(a,b)> = <(x,y),T(a,b)>

where <,> denotes the inner product on C^2.

To find T*, we need to compute <T*(x,y),(a,b)> for any (x,y) and (a,b) in C^2. We have

<T*(x,y),(a,b)> = <(x,y),T(a,b)>

= <(x,y),(2a+ib,(1-2)a)>

= 2ax + ibx + (1-2)ay

= (2a-2y)x + ibx

Thus, we see that T*(x,y) = (2x-2y, ix) for any (x,y) in C^2.

Now, to compute T*(3i,1+2i), we have

T*(3i,1+2i) = (2(3i)-2(1+2i), i(3i))

= (6i-2-4i, -3)

= (4i-2, -3)

Therefore, T*(3i,1+2i) = (4i-2, -3).

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The given question is incomplete, the complete question is:

Consider the linear operator T on C^2 defined by T(a, b) = (2a + ib, (1-2)a) Compute T*(3 i,1 2i)

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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Answer:n=sqrt55

Step-by-step explanation:

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?

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

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.

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.

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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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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Answer:

30

because 1=3 so each of them are 30

The total amount that she pays for the plastic rod is $1.925

We know that the cost of the plastic rod is $0.35 per foot, including the tax.

We also know that Maria buys (5 + 1/2) feet of the plastic rood, then to find the total cost we only need to take the product between the cost per foot and the number of feet that she buys.

We will get:

total cost = (5 + 1/2)*$0.35

total cost = 5.5*$0.35

total cost = $1.925

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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 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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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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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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Answer:

Step-by-step explanation:

1. The volume of the prism is 144 cubic units

2. The area of the shaded base is 16units²

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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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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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.

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

The angle of elevation is a function of the height of the space shuttle given by θ = arctan(h / 2).

(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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The 72nd percentile of the normally distributed random variable x is 196.788, rounded to the tenths place.

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.

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 measure of the inscribed angle is determined as 150⁰.

The perimeter of the circle is calculated as follows;

P = 2πr

where;

P = 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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