Applied Mathematics

Calculus

Medium

applied

If the total cost and total revenue of a company that produces and sells $x$ units of a particular product are $C(x) = 5x + 350$ and $R(x) = 50x - x^2$ respectively, then which of the following is/are the breakeven values:

(A) $x = 10$

(B) $x = 25$

(C) $x = 45$

(D) $x = 30$

Choose the correct answer from the options given below:

(A) and (D) only

(A), (B) and (C) only

(A) only

✅ Correct Option: 4

Breakeven occurs when Total Cost equals Total Revenue.

Given:

Cost function: $C(x) = 5x + 350$

Revenue function: $R(x) = 50x - x^2$

At breakeven:

$C(x) = R(x)$

$5x + 350 = 50x - x^2$

$x^2 + 5x + 350 - 50x = 0$

$x^2 - 45x + 350 = 0$

Factoring the quadratic equation:

Two numbers that multiply to give 350 and add to give -45 are -35 and -10.

$(x - 35)(x - 10) = 0$

$x = 35$ or $x = 10$

The breakeven values are $x = 10$ and $x = 35$ .

Checking the given options:

(A) $x = 10$ ✓

(B) $x = 25$ ✗

(D) $x = 30$ ✗

Only option (A) $x = 10$ is a breakeven value among the choices given.

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Solution