1. A company that manufactures thingamajigs has a revenue function, R(x)=−2x^2+36x−90 , that calculates the companies revenue (R) in thousands of dollars if they make x thousand thingamajigs. They also have a cost function, C(x)=2x+14 , that calculates the cost (C) in thousands of dollars if they make x thousand thingamajigs. Determine the numbers of thingamajigs that will produce a profit for the company. Note that profit only occurs when R>C
A. The company will make a profit if they manufacture
B. The company will make a profit if they manufacture between 4 and 13 thousand thingamajigs.
C. There is no way for the company to make a profit if they manufacture 8,500 thingamajigs.
D. The company will make a profit if they manufacture between 3 and 15 thousand thingamajigs
2. A bottle rocket starts at a height of 8 feet above the ground before it launches. It reaches it's maximum height of 80 feet 3 seconds after launch and begins to fall down. Write a quadratic function to model the height (h) of the rocket given the time (t) since launch.
A. h(t)=-8(t-3)^2+80
B. h(t)=8(t+3)^2-80
C. h(t)=8(t-3)^2-80
D. h(t)=-8(t+3)^2+80
3. A company has a model for their profit (p) given they produce x gizmos. They have lost the model, but someone remembers the model said they would make no profit if they produced 3 or 9 thousand gizmos. Another person remembered that the model said they would maximize their profit if they made 6 thousand gizmos, and that it would be 63 thousand dollars of profit. Using this information, write a quadratic function that can mimic their original model. Where p is the profit in thousands of dollars and x is the number of gizmos produced in the thousands.
A. p(x)=−7(x+3)(x+9)
B. p(x)=−7(x−3)(x−9)
C. p(x)=7/15(x+3)(x+9)
D. p(x)=−7(x+6)^2+63
4. Calculate the Average Rate of Change of the function represented in the table below for the interval x=−1 to x=2 .
A. -1
B. 3
C. 1
D. -3
5. Below is the graph of a system of equations. Determine the solution(s) to that system of equations.
A. (−1,−2) and (4, 3)
B. (−1,−2) and (4, 3)
C. (−2,0), (1, 0) and (3, 0)
D. (0,−1) and (0,−3)
Since there are multiple questions in this prompt, I will answer each question one by one.
1. To determine the number of thingamajigs that will produce a profit for the company, we need to find the values of x for which R(x) is greater than C(x). Substitute the revenue function and cost function into the inequality R(x) > C(x):
-2x^2 + 36x - 90 > 2x + 14
Simplifying the inequality:
-2x^2 + 34x - 104 > 0
To solve this inequality, we can first find the values of x for which the quadratic equation is equal to zero:
-2x^2 + 34x - 104 = 0
Using the quadratic formula, we find:
x = (-34 ± sqrt(34^2 - 4(-2)(-104))) / (2(-2))
Simplifying further:
x = (-34 ± sqrt(1156-832)) / (-4)
x = (-34 ± sqrt(324)) / -4
x = (-34 ± 18) / -4
Therefore, x = (-34 + 18)/-4 or x = (-34 - 18)/-4
x = -16/ -4 or x = -52/ -4
x = 4 or x = 13
Therefore, the company will make a profit if they manufacture between 4 and 13 thousand thingamajigs. Answer choice B is correct.
2. Since the rocket starts at a height of 8 feet and reaches a maximum height of 80 feet after 3 seconds, we can determine the quadratic function by using the vertex form of a quadratic equation, which is h(t) = a(t - h)^2 + k.
The vertex of the parabola is (3, 80), and since the rocket starts at a height of 8 feet, we can substitute these values into the vertex form equation to find the value of a:
80 = a(3 - 3)^2 + 8
80 = a(0) + 8
80 = 8
a = 8/0
a is undefined.
Since a is undefined, the correct option is not listed. None of the answer choices accurately represents the quadratic function for the height of the rocket.
3. To determine the quadratic function that can mimic the original model, we need to consider the given information. The profit is zero when x = 3 and x = 9, so we know that the quadratic function has roots at x = 3 and x = 9. Additionally, when x = 6, the profit is maximized at $63,000.
Using these pieces of information, we can write the quadratic function as:
p(x) = a(x - 3)(x - 9)
To determine the value of a, we substitute the coordinates (6, 63) into the quadratic function:
63 = a(6 - 3)(6 - 9)
63 = -3a(3)
63 = -9a
a = -7
Substituting this value of a back into the quadratic function:
p(x) = -7(x - 3)(x - 9)
Therefore, answer choice A, p(x) = -7(x + 3)(x + 9), is correct.
4. To calculate the average rate of change of a function over an interval, we use the formula:
Average rate of change = (f(b) - f(a))/(b - a)
In this case, a = -1 and b = 2. The table is not provided, so we cannot calculate the average rate of change without the values of the function at x = -1 and x = 2. Without additional information, none of the answer choices can be determined.
5. Since the graph of the system of equations is not provided, we cannot determine the solution(s) without knowing the equations. None of the answer choices can be determined without additional information.
In summary, we can determine that the company will make a profit if they manufacture between 4 and 13 thousand thingamajigs (Answer choice B) and that the quadratic function that mimics the original model is p(x) = -7(x + 3)(x + 9) (Answer choice A). However, we cannot determine the average rate of change or the solution(s) to the system of equations without additional information.