Question

find two positive numbers that satisfy the given requirements. (enter your answers as a comma - separated list.) the sum of the first and twice the second is 360 and the product is a maximum.

Explanation:

Step1: Define variables

Let the first number be \(x\) and the second number be \(y\). We know \(x + 2y=360\), so \(x = 360 - 2y\). The product \(P=xy=y(360 - 2y)=360y-2y^{2}\).

Step2: Find the derivative of the product function

Differentiate \(P(y)\) with respect to \(y\). Using the power rule \((ax^{n})^\prime=anx^{n - 1}\), \(P^\prime(y)=\frac{d}{dy}(360y-2y^{2})=360-4y\).

Step3: Find the critical points

Set \(P^\prime(y) = 0\), then \(360-4y = 0\). Solving for \(y\):

$$
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$$

Step4: Check the second - derivative

Differentiate \(P^\prime(y)\) to get \(P^{\prime\prime}(y)=-4<0\). Since \(P^{\prime\prime}(y)<0\) when \(y = 90\), the function \(P(y)\) has a maximum at \(y = 90\).

Step5: Find the value of \(x\)

Substitute \(y = 90\) into \(x=360 - 2y\). Then \(x=360-2\times90=180\).

Answer:

\(180,90\)