A small company has the marketing information that 35 units will sell daily at a price of $34.75 per unit, and that sales will rise to 36 units per day at a price of $33.06 per unit. Use this information to create a linear demand function, then create the associated revenue function and find the price that will yield the maximum revenue.

a) The derivative f'(x) is 12x².

b) The amount of f'(1) is 12.

c) The equation of the tangent line is 12x - 7.

a) To find the derivative of f(x), we can use the power rule for each term. The power rule states that for any function f(x) = xⁿ, the derivative f'(x) = nx^(n-1). So for f(x) = 4x³ + 1, Thus, the derivative f'(x) = 12x².

b) To compute the derivative of f at x=1, we simply plug in x=1 into the derivative equation we found in part a. So f'(1) = 12(1)² = 12.

c) To determine the equation of the tangent line to the curve y=f(x) at the point (1,5), we use the point-slope form of a line, which is y-y1 = m(x-x1), where m is the slope of the line and (x1,y1) is the point the line passes through. The slope of the tangent line is the derivative of f at x=1, which we found in part b to be 12. So the equation of the tangent line is y-5 = 12(x-1), or y = 12x - 7.

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The time it takes for the bacteria to reach 1,000,000 is given as follows:

t = 164.5 minutes.

The number of bacteria after t minutes is given by the exponential function presented as follows:

P(t) = 1000e^(0.042t).

The population will reach 1,000,000 when:

P(t) = 1,000,000.

Hence the time needed is obtained solving the equation as follows:

1,000,000 = 1000e^(0.042t).

e^(0.042t) = 1,000,000/1,000

e^(0.042t) = 1000.

The inverse operation regarding the exponent e is the natural logarithm, hence we can isolate t as follows:

0.042t = ln(1000)

t = ln(1000)/0.042

t = 164.5 minutes.

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In the given figure the equation m∠MRS + m∠MRO = 180° is the postulate describing the linear pair of angles.

The English word "angle" derives from the Latin word "angulus," which means "corner." The vertex and the two rays are referred to as the sides of an angle, respectively, and are the shared termini of two rays.

It is given that -

∠MRS ≅ ∠MRO

Now, what that means is that ∠MRS is congruent to ∠MRO.

Thus, to understand this concept of equality or congruence, we can prove it since from the given image, we see that -

m∠MRS = m∠MRO = 90°

Since they are both right angles then we can say that the correct theorem is the definition of a right angle triangle.

m∠MRS + m∠MRO = 180°

It is seen that the angles form a linear pair and lie on the same plane.

Therefore, the correct postulate is linear pair.

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For 6 cities, the number of roads would be 9. This is because each city must be connected to every other city, and there are 5 pairs of cities that need to be connected, which means 5 roads.

Roads are a form of transportation infrastructure consisting of a prepared surface for vehicles to travel on. They are typically made from asphalt or concrete and are designed to provide a safe and efficient means of travel.

Then, the remaining 4 cities need to be connected, which means 4 more roads for a total of 9.
For 7 cities, the number of roads would be 12. This is because each city must be connected to every other city, and there are 6 pairs of cities that need to be connected, which means 6 roads. Then, the remaining 5 cities need to be connected, which means 5 more roads for a total of 12.
For N cities, the number of roads would be equal to N(N-1)/2. This is because each city must be connected to every other city, and there are N(N-1)/2 pairs of cities that need to be connected. For example, if N = 10, then 10 × 9 = 90 pairs of cities need to be connected, which means 90 roads.

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Rounding to the nearest percent, we get that the percent change in the number of migrating cranes from the first March to the next is approximately -50%.

To calculate the percentage, we must first divide the amount by the total value and then multiply the result by 100.

To find the percent change in the number of migrating cranes from the first March to the next, we can use the formula:

percent change = ((new value - old value) / old value) × 100%

where the old value is the number of migrating cranes in the first March and the new value is the number of migrating cranes in the next March.

Substituting the given values, we get:

percent change = ((3480 - 6955) / 6955) × 100%

Simplifying, we get:

\percent change = (-3475 / 6955) × 100%

percent change ≈ -50%

Rounding to the nearest percent, we get that the percent change in the number of migrating cranes from the first March to the next is approximately -50%.

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The number of $0.49$ stamps Paula bought is s = (1274 - 49s / 21) + 6.

To solve the equation 0.49s + 0.21(s - 6) = 12.74 for s, we can multiply both sides by 100 to get:

49s + 21(s - 6) = 1274

Now, we can move the 21(s - 6) term to the left side of the equation and the 49s term to the right side of the equation. To do this, we will subtract the 49s from both sides of the equation, which yields:

21(s - 6) = 1274 - 49s

Next, we can divide both sides of the equation by 21 to isolate the (s - 6) term, resulting in:

(s - 6) = 1274 - 49s / 21

Finally, we can add 6 to both sides of the equation to solve for the number of $0.49$ stamps that Paula bought:

s = (1274 - 49s / 21) + 6

Therefore, the number of $0.49$ stamps Paula bought is s = (1274 - 49s / 21) + 6.

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Using geometric sequences, 20,480 rabbits are grown during the 12th month.

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the preceding term by a fixed non-zero number called the common ratio.

At the end of the first month, there are 10 x 2 = 20 rabbits.

At the end of the second month, there are 20 x 2 = 40 rabbits.

Similarly, at the end of the third month, there are 40 x 2 = 80 rabbits.

This is a geometric sequence with a common ratio of 2 and a first term of 10. The number of rabbits at the end of the 12th month would be:

10 x 2^12 = 10 x 4096 = 40,960 rabbits

To find the number of rabbits grown during the 12th month, we need to subtract the number of rabbits at the end of the 11th month from the number of rabbits at the end of the 12th month:

[tex]40,960 - (10 x 2^{11}) = 40,960 - 20,480 = 20,480[/tex]

Hence, 20,480 rabbits are grown during the 12th month.

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So the fourth force FA that would need to be added to make the net force zero is < 6, 9 >.

The net force on an object is the sum of all the forces acting on it. In this case, there are three different forces acting on the object: F1, F2, and F3. Each of these forces has a magnitude of < -2, -3 >. To find the net force Fnet, we simply add up all the forces:

Fnet = F1 + F2 + F3
Fnet = < -2, -3 > + < -2, -3 > + < -2, -3 >
Fnet = < -6, -9 >

To find the fourth force FA that would need to be added to make the net force zero, we simply need to find a force that is equal and opposite to Fnet. That is:

FA = -Fnet
FA = -< -6, -9 >
FA = < 6, 9 >

So the fourth force FA that would need to be added to make the net force zero is < 6, 9 >.

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50 possible combinations

The probability that both of the rolls are made by the same participant is 25/99, since there are 25 participants and the second roll must be chosen from the remaining 99 rolls. The probability that both rolls have the same protein is 4/99, since there are 4 types of proteins and the second roll must be chosen from the remaining 99 rolls. Lastly, the probability that one roll has white rice and one roll has brown rice is 50/99, since there are 50 possible combinations of white and brown rice and the second roll must be chosen from the remaining 99 rolls.

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There is no solution for the equation   -6-(4x)/(x+2)=(8)/(x+2).

To solve for all values of x in the equation -6-(4x)/(x+2)=(8)/(x+2), we need to use the following steps:

Step 1: Multiply both sides of the equation by (x+2) to eliminate the denominators. This gives us:
-6(x+2) - 4x = 8

Step 2: Simplify both sides of the equation by distributing and combining like terms. This gives us:
-6x - 12 - 4x = 8
-10x - 12 = 8

Step 3: Add 12 to both sides of the equation to isolate the variable on one side. This gives us:
-10x = 20

Step 4: Divide both sides of the equation by -10 to solve for x. This gives us:
x = -2

However, we need to check our solution to make sure it is not an extraneous solution. If we plug x = -2 back into the original equation, we get:
-6 - (4(-2))/(-2+2) = (8)/(-2+2)
-6 - (-8)/0 = (8)/0

Since we cannot divide by zero, our solution of x = -2 is an extraneous solution and there are no valid solutions to this equation. Therefore, the answer is no solution.

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Answer:

To create a perfect-square binomial of the form (y - k)^2, we need to find the value of k such that:

the first term of the binomial is y^2 (which is already the case)

the second term of the binomial is -2ky (which corresponds to -36y in the given expression)

the third term of the binomial is k^2 (which corresponds to 81 in the given expression)

To find k, we can use the formula:

k = (1/2)*(-b/a)

where a is the coefficient of y^2, b is the coefficient of y, and we are looking for the value of k that makes the expression a perfect square.

In this case, a = 1 and b = -36, so:

k = (1/2)(-b/a) = (1/2)(-(-36)/1) = 18

Therefore, the missing number to create a perfect-square binomial is 18:

(y - 18)^2 = y^2 - 36y + 324

The constant proportionality between the x-coordinates and y-coordinates of these four points is 12.

What is Proportionality?

Proportionality refers to the relationship between two quantities that change in a consistent and predictable way. In a proportional relationship, when one quantity changes, the other quantity changes in a constant ratio or proportion to the first.

Assuming that you want to find the constant proportionality between the x-coordinates and y-coordinates of these four points:

First, we can calculate the ratios of the y-coordinates to the x-coordinates for each point:

For (3,36): y/x = 36/3 = 12

For (5,60): y/x = 60/5 = 12

For (6,72): y/x = 72/6 = 12

For (7,84): y/x = 84/7 = 12

We can see that the ratios are all equal to 12. Therefore, the constant proportionality between the x-coordinates and y-coordinates of these four points is 12.

Note that if you had asked for the constant proportionality between the differences in the y-coordinates and the differences in the x-coordinates of these points, the answer would be different. That constant proportionality is known as the slope of the line passing through these points.

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The correct answer is "No, because for each input there is not exactly one output".

A function is a mathematical rule that assigns a unique output value for each input value. It is a set of ordered pairs where the first element is the input and the second element is the output.

The relation is not a function because for the input value of 1, there are two possible output values: 0 and 2. In a function, each input can have only one output value. However, in this relation, the input value of 1 is associated with two different output values, so it violates the definition of a function. Therefore, the correct answer is "No, because for each input there is not exactly one output".

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The polynomial in factored form is: (x-5)(x+6)(x-(3-2i))(x-(3+2i))

Since complex roots always come in conjugate pairs, we know that if 3-2i is a root, then 3+2i must also be a root. Therefore, we can write the polynomial in factored form by multiplying together the factors (x-5), (x+6), (x-(3-2i)), and (x-(3+2i)).

Alternatively, we can write the polynomial in factored form using the formula for a polynomial with complex roots:

p(x) = (x-r1)(x-r2)(x-r3)(x-r4)

where r1, r2, r3, and r4 are the roots of the polynomial. In this case, r1 = 5, r2 = -6, r3 = 3-2i, and r4 = 3+2i. Substituting these values into the formula gives us:

p(x) = (x-5)(x+6)(x-(3-2i))(x-(3+2i))

Therefore, the polynomial in factored form is written in the following form: (x-5)(x+6)(x-(3-2i))(x-(3+2i)).

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Answer: 50 total stamps = 30 Postcard Stamps + 20 Letter Stamps

Step-by-step explanation:

Postcard Stamp = Ps

Letter Stamp = Ls

Ps = 20c

Ls = 33c

50 total stamps = $12.60

50 total stamps = X Ps + Y Ls

12.60 = 30 Ps + 20 Ls

$12.60 = 30(20c) + 20(33c)

50 total stamps = 30 Postcard Stamps + 20 Letter Stamps

Answer:

You have 30 postcard stamps and 20 letter stamps.

Step-by-step explanation:

To solve this you'll need to set up a system of equations. Let's use P for postcard stamps and L for letter stamps

Remember, since the price is in cents, it'll be 0.2 and 0.33.

The equations can be in any order, this is just the order I chose.

0.2P + 0.33L = 12.60

P + L = 50

Your next step is to cancel out one of the variables to solve for the other. Let's cancel out P and solve for L. (You can switch these if you want, you'll still get the same answsrs.) Remember to multiply by a negative so the variable cancels out.

Here's what your work will look like:

0.2P + 0.33L = 12.60

-0.2(P + L = 50)

Here's what your new equations will look like after distributing:

0.2P + 0.33L = 12.60

-0.2P - 0.2L = -10

Now add these two equations together. When you do so, the P cancels out, and you can now solve for L.

Here's what your new equation will look like after adding the two equations:

0.13L = 2.6

Now, divide both sides by 0.13 to get what L equals. After doing so, you should get L = 20. This means that you have 20 letter stamps.

Your last step is to plug the value of L (which is 20) into either of your original equations to solve for how many postcard stamps you have. Let's use our second equation, P + L = 50. (You can use either original equation and get the same answer, but this one is more simpler to use.)

Here's what your work will look like:

P + 20 = 50

Subtract 20 from both sides:

P = 30. This means you have 30 postcard stamps.

Hope this helps!

The equation that can be used to determine the number of days is $123 = $55 + $8.50d

An equation is an expression that has an equal to sign. A linear equation is an equation that has a single variable raised to the power of one. The form of a linear equation is:

y = mx + b

Where:

The form of the linear equation that can be used to determine the number of days is:

Total cost = flat fee + (cost per day x number of days)

$123 = $55 + ($8.50 x d)

$123 = $55 + $8.50d

d = ($123 - $55) / 8.50

d = 8 days

Here is the complete question:

Val rented a bicycle while she was on vacation. She paid a flat rental fee of $55.00, plus $8.50 each day. The total cost was $123. Write an equation you can use to find the number of days she rented the bicycle.

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The volume of the rectangular pyramid is 1200 m³, the volume of the oblique cone is 5890 ft³ while the volume of the triangular prism is 321.75 ft³

An equation is an expression that shows the relationship between numbers and variables using mathematical operations like exponents, addition, subtraction, multiplication and division.

a) The volume of the rectangular pyramid is:

Volume = (1/3) * area of base * height

Substituting:

Volume = (1/3) * (15 m * 12 m) * 20 m = 1200 m³

b) The volume of the oblique cone is:

Volume = (1/3) * π * radius² * height

but radius = 30 ft / 2 = 15 ft,

Substituting:

Volume = (1/3) * π * 15² * 25  = 5890 ft³

The volume of the cone is 5890 ft³

c) The volume of the triangular prism is:

Volume = area of base * height

Substituting:

Volume = (1/2 * 9 ft * 5 ft) * 14.3 ft = 321.75 ft³

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The solutions for θ in the interval [0, 2π) where cos(θ) = -√3/2 are θ = 2π/3 and θ = 4π/3.

The graph of f(θ) is shifted and stretched when compared to the graph of f(2θ).

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and the trigonometric functions that describe those relationships.

Part A:

To find when the pogo stick's spring will be equal to its non-compressed length, we need to solve for when f(θ) = 0.

f(θ) = 2cos(θ) + √3 = 0

2cos(θ) = -√3

cos(θ) = -√3/2

The solutions for θ in the interval [0, 2π) where cos(θ) = -√3/2 are θ = 2π/3 and θ = 4π/3.

Part B:

If we double the angle, θ becomes 2θ, and the function becomes:

f(2θ) = 2cos(2θ) + √3

Using the double angle formula for cosine, we can rewrite this as:

f(2θ) = 2(2cos²(θ) - 1) + √3

f(2θ) = 4cos²(θ) - 2 + √3

Substituting cos(θ) = -√3/2, we get:

f(2θ) = 4(-3/4) - 2 + √3

f(2θ) = -3 + √3

So the solutions for 2θ in the interval [0, 2π) where f(2θ) = 0 are:

2θ = π/6 and 2θ = 11π/6

Dividing by 2, we get the solutions for θ:

θ = π/12 and θ = 11π/12

These solutions are different from the solutions in Part A, and the graph of f(θ) is shifted and stretched when compared to the graph of f(2θ).

Part C:

To find when the lengths of the springs are equal, we need to solve the equation f(θ) = g(θ).

2cos(θ) + √3 = 1 - sin²(θ) + √3

2cos(θ) = 1 - sin²(θ)

Using the identity sin²(θ) + cos²(θ) = 1, we can rewrite this as:

2cos(θ) = cos²(θ)

cos(θ)(cos(θ) - 2) = 0

The solutions for θ in the interval [0, 2π) where the lengths of the springs are equal are:

θ = 0, θ = π/3, θ = 2π/3, θ = π, θ = 4π/3, θ = 5π/3

We can check that f(θ) = g(θ) at each of these values.

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Answer: A = 1/2(8+12)h

Step-by-step explanation: The formula for area of a trapezoid is 1/2(b1+b2)h so if you just sub in all the variables "a" would be correct

An equation for Area of trapezoid is,

A = (8 + 12) h / 2

We have,

The bases of a trapezoid are 8 centimeters and 12 centimeters, and the height is h centimeters.

We know that,

Area of trapezoid is,

A = (a + b) h / 2

Where, a and b are bases and h is height of trapezoid.

Here,, a = 8 cm,

b = 12 cm

Hence, Area of trapezoid is,

A = (8 + 12) h / 2

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The point (a, b) in QII that is also on the unit circle is (-84/85, 13/85).

So, (a,b) = (-84/85, 13/85).

Since the point (84/85, -13/85) is on the unit circle with the center at the origin, the point (a, b) in QII is also on the circle if it satisfies the equation of the unit circle centered at the origin, which is x^2 + y^2 = 1.

In QII, the x-coordinate is negative and the y-coordinate is positive. So, we can write (a, b) as (-a, b), where a > 0 and b > 0.

Substituting (-a, b) into the equation of the unit circle, we get:

(-a)^2 + b^2 = 1

a^2 + b^2 = 1

Since (84/85, -13/85) is on the unit circle, we can use the Pythagorean theorem to find the value of a and b:

(84/85)^2 + (-13/85)^2 = 1

a^2 = (84/85)^2 = 7056/7225

b^2 = (-13/85)^2 = 169/7225

Taking the square root of both sides, we get:

a = sqrt(7056/7225) = 84/85

b = sqrt(169/7225) = 13/85

Therefore, the point (a, b) in QII that is also on the unit circle is (-84/85, 13/85).

So, (a,b) = (-84/85, 13/85).

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Answer:

0.40 or 40% is shaded

Step-by-step explanation:

There are 40 shaded out of the 100 squares. Therefor, .40 is the answer

Jada has earned a total of 400 points so far.

    141 + 87 + 81 + 91 = 400

There are a total of 450 points possible so far.

    150 + 100 + 100 + 100 = 450

Jada has 400/450 or about 88.89% of the possible points, so no, she does not have 90%.

Adding in a 100-point test, the total number of points would become 550 and 90% of 550 is 495 points.

    0.90 x 550 = 495

To finish the class with an A, Jada needs to have 495 points.  She currently has 400 points.  This means she needs a 95 on the final to finish the class with an A.

The inequality that represents how many DVDs Andy can rent in one month that satisfies this condition is 10 + .75n ≤ 25

The correct answer choice is option B

Cost of rental per month = $10

Additional cost = $0.75

Number of DVD's rented = n

Total amount Andy want to spend ≤ $25

The inequality:

10 + 0.75n ≤ 25

subtract 10 from both sides

0.75n ≤ 25 - 10

0.75n ≤ 15

divide both sides by 0.75

n ≤ 15 / 0.75

n ≤ 20

Ultimately, Andy can rent no more than 20 DVD's in a month.

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There are a total of 28 different 3-digit numbers that can be formed from the numbers 1,3,5,7,9 if each number formed is greater than 300. This can be found by using the formula for distinguishable permutations.

There are a total of 5 different 3-digit numbers that can be formed from the numbers 1,3,5,7,9 if each number formed is greater than 300. These numbers are 315, 357, 375, 397, and 359. This can be found by using the formula for distinguishable permutations, which is n!/(n-r)!, where n is the total number of items and r is the number of items chosen. In this case, n is 5 (the numbers 1,3,5,7,9) and r is 3 (the number of digits in each number formed).

The formula for distinguishable permutations is:

n!/(n-r)!

= 5!/(5-3)!

= 120/2

= 60

Therefore, there are a total of 60 different 3-digit numbers that can be formed from the number 1,3,5,7,9. However, we need to subtract the numbers that are less than 300, which are the numbers formed from the digits 1 and 3 in the hundreds place. There are 2 numbers in the hundreds place (1 and 3) and 4 numbers in the tens and ones place (3,5,7,9), so there are a total of 2*4*4 = 32 numbers less than 300. Subtracting these from the total gives us:

60 - 32 = 28

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The measure of angle 2 is 70° and the measure of angle 3 is also 70°°.

We know when a transversal intersects two parallel lines at two distinct points,

Two pairs of interior and alternate angles are formed such that the measure of interior angles are same and the measure of alternate angles is also the same.

From the given information ∠1 and ∠3 are pairs of alternate interior angles.

Therefore, m∠1 = m∠B = 70°.

We also know that the vertically opposite angles are equal,

Here ∠1 and ∠2 are vertically opposite angles,

Therefore, m∠1 = m∠2 = 70°.

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Using synthetic division, h(2) = -9.

To find h(2) using synthetic division, we will divide the polynomial h(z) by (z-2). The steps are as follows:

1. Write the coefficients of the polynomial in a row: 2 -10 20 -25 3
2. Write the value of z we are dividing by in the upper left corner: 2 | 2 -10 20 -25 3
3. Bring down the first coefficient: 2 | 2 -10 20 -25 3
      -------------------
          2
4. Multiply the first coefficient by the value of z and write the result below the second coefficient: 2 | 2 -10 20 -25 3
      -------------------
          2   4
5. Add the second coefficient and the result from step 4: 2 | 2 -10 20 -25 3
      -------------------
          2  -6
6. Repeat steps 4 and 5 for the remaining coefficients: 2 | 2 -10 20 -25 3
      -------------------
          2  -6  8  -6
7. The last number in the bottom row is the remainder: 2 | 2 -10 20 -25 3
      -------------------
          2  -6  8  -6  -9
8. The value of h(2) is the remainder, so h(2) = -9.

Therefore, the answer is h(2) = -9.

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The gravitational pull of the Sun on the object would decrease as the distance between them increased.

If the velocity of an orbiting body were increased, its orbital path would change from an elliptical orbit to a parabolic or hyperbolic orbit, depending on the extent of the increase in velocity.

The amount of gravitational pull on the object would decrease as the distance between the object and the Sun increased. This is because gravitational force decreases with distance according to the inverse-square law.

In summary, if the velocity of an orbiting body were increased to the point where its orbital path changed into a parabolic or hyperbolic orbit, the object would eventually escape the Sun's gravitational pull and move away from it in a straight line.

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