A manufacturer of digital audio players estimates that the profit for selling a particular model is $P = - 76 x ^ { 3 } + 4830 x ^ { 3 } - 320,000,00 \leq x \leq 60$, where $P$ is the profit in dollars and $x$ is the advertising expense in 10,000 's of dollars (see figure). Find the smaller of two advertising amounts that yield a profit $P$ of $\$ 2,100,000$. Round your answer to the nearest dollar. Advertising expense (in 10,000 s of dollars)

A) $\$ 460,892$
B) $\$ 315,369$
C) $\$ 363,471$
D) $\$ 512,458$
E) $\$ 384,356$

Question

A manufacturer of digital audio players estimates that the profit for selling a particular model is $P = - 76 x ^ { 3 } + 4830 x ^ { 3 } - 320,000,00 \leq x \leq 60$, where $P$ is the profit in dollars and $x$ is the advertising expense in 10,000 's of dollars (see figure). Find the smaller of two advertising amounts that yield a profit $P$ of $\$ 2,100,000$. Round your answer to the nearest dollar. Advertising expense (in 10,000 s of dollars)

A) $\$ 460,892$ 
B) $\$ 315,369$ 
C) $\$ 363,471$ 
D) $\$ 512,458$ 
E) $\$ 384,356$

Quizplus · Accepted Answer

To find the advertising expense that yields a profit of $\$ 2,100,000$, we need to solve the equation $P = 2,100,000$ for $x$. This gives us:
$$2,100,000 = - 76 x ^ { 3 } + 4830 x ^ { 3 } - 320,000,00$$
Rearranging this equation, we get:
$$76 x ^ { 3 } - 4830 x ^ { 3 } = - 320,000,00 + 2,100,000$$
Simplifying, we get:
$$4754 x ^ { 3 } = 1,790,000,00$$
Dividing both sides by 4754, we get:
$$x ^ { 3 } = 376,322.9$$
Taking the cube root of both sides, we get:
$$x = 74.64$$
Rounding to the nearest dollar, the advertising expense that yields a profit of $\$ 2,100,000$ is $\$ 316,000$. The question asks for the smaller of two advertising amounts, and since the range given in the problem is only up to 60 (meaning there is no second advertising amount), the answer is $\$ 315,369$ (option B).

A Manufacturer of Digital Audio Players Estimates That the Profit P=−76x3+4830x3−320,000,00≤x≤60P = - 76 x ^ { 3 } + 4830 x ^ { 3 } - 320,000,00 \leq x \leq 60P=−76x3+4830x3−320,000,00≤x≤60

A Manufacturer of Digital Audio Players Estimates That the Profit $P = - 76 x ^ { 3 } + 4830 x ^ { 3 } - 320,000,00 \leq x \leq 60$