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

Question

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

Answer 1

To determine the numbers of thingamajigs that will produce a profit for the company, we need to find the values of x for which R(x) > C(x).

Given:
R(x) = -2x^2 + 36x - 90
C(x) = 2x + 14

Setting R(x) > C(x), we have:
-2x^2 + 36x - 90 > 2x + 14

Rearranging the inequality, we get:
-2x^2 + 36x - 90 - 2x - 14 > 0
-2x^2 + 34x - 104 > 0

Now, we can solve this quadratic inequality by factoring or using the quadratic formula. However, it is easier to solve by factoring since the leading coefficient is -2.

Factoring the quadratic, we have:
-2(x^2 - 17x + 52) > 0

The quadratic factors as:
-2(x - 4)(x - 13) > 0

Now, we can set each factor equal to zero to determine the critical points:
x - 4 = 0 => x = 4
x - 13 = 0 => x = 13

The critical points divide the number line into three intervals: (-∞, 4), (4, 13), and (13, ∞).
We need to test a value within each interval to determine the sign of the quadratic inequality.

For x < 4, let's test x = 0:
-2(0 - 4)(0 - 13) > 0
-2(4)(-13) > 0
104 > 0
Since the inequality is true for x < 4, we include this interval in the solution.

For 4 < x < 13, let's test x = 10:
-2(10 - 4)(10 - 13) > 0
-2(6)(-3) > 0
36 > 0
Since the inequality is true for 4 < x < 13, we include this interval in the solution.

For x > 13, let's test x = 15:
-2(15 - 4)(15 - 13) > 0
-2(11)(2) > 0
-44 > 0
Since the inequality is false for x > 13, we exclude this interval from the solution.

Therefore, the numbers of thingamajigs that will produce a profit for the company are between 4 and 13 thousand thingamajigs.
B. The company will make a profit if they manufacture between 4 and 13 thousand thingamajigs.