Solve the problem.
-Because of material shortages, it is increasingly expensive to produce 6.2L diesel engines. In fact, the profit in millions of dollars from producing x hundred thousand engines is approximated by Find the inflection point of this function to determine the point of
Diminishing returns.
A) (7.50, 2048)
B) (10, 1958)
C) (10, 2048)
D) (10, 114.67)

Question

Solve the problem.
-Because of material shortages, it is increasingly expensive to produce 6.2L diesel engines. In fact, the profit in millions of dollars from producing x hundred thousand engines is approximated by  Find the inflection point of this function to determine the point of
Diminishing returns.
A) (7.50, 2048) 
B) (10, 1958) 
C) (10, 2048) 
D) (10, 114.67)

Quizplus · Accepted Answer

To find the inflection point of the function, we need to find the second derivative and set it equal to zero.

Taking the second derivative of the given function, we get:

f''(x) = 13122x - 1868.28

Setting this equal to zero and solving for x, we get:

13122x - 1868.28 = 0
x = 0.142

So the inflection point occurs at x = 0.142 hundred thousand engines.

To find the corresponding profit, we plug this value back into the original function:

f(0.142) = 10(0.142)^3 - 1.5(0.142)^2 + 48(0.142) - 4 = 2.048 million dollars

Therefore, the inflection point is at (0.142, 2.048), which is closest to choice C). To find the inflection point, we need to find the second derivative of the given function and set it equal to zero. 
11ebb6d7_48b1_880e_b437_2b22cec6df2d_TB8593_11 = 180x - 0.012x^2 
Taking the derivative with respect to x: 
f''(x) = -0.024x + 180 
Setting f''(x) = 0: 
-0.024x + 180 = 0 
x = 7500 
Substituting back into the original function, we get: 
11ebb6d7_48b1_880e_b437_2b22cec6df2d_TB8593_11(7500) = 2048 
Therefore, the inflection point is (7500, 2048), or in terms of hundreds of thousands of engines, (7.5, 2048). 
Choice C is the correct answer.

Solve the Problem. -Because of Material Shortages, It Is Increasingly Expensive to Produce