Question

The marginal revenue (in thousands of dollars) from the sale of x gadgets is given by the following function:

R′(x) = 4x(x²+ 26,000)⁻²/³

(a) Find the total revenue function if the revenue from 125 gadgets is $17,939.

(b) How many gadgets must be sold for a revenue of at least $45,000?

Answer 1

Answer:

  (a)  R(x) = 6∛(x² +26000) -190

  (b)  185 or more

Step-by-step explanation:

(a) Given R(125) = 17.939, we can write the total revenue function as ...

$R(x)=\displaystyle2\int\limits^x_{125} {(x^2+26000)^{-\frac{2}{3}}} \, 2x\cdot dx+17.939 =6\sqrt[3]{x^2+26000}-207.939+17.939\\\\R(x)=6\sqrt[3]{x^2+26000}-190$

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(b) We want to find x for R(x) = 45. Then ...

  45 = 6∛(x² +26000) -190

  235/6 = ∛(x² +26000) . . . . . . add 190; divide by 6

  60082.75 -26000 = x² . . . . . cube both sides; subtract 26000

  √34082.75 ≈ x ≈ 184.6

At least 185 gadgets must be sold for a revenue of at least $45,000.