Question

a companys profit g (in thousands of dollars) after a hours of operation is modeled by the equation: $g = \sqrt4{40a - 319} - 1$ find the time when the profit will reach 2 thousand dollars. show your work here

Explanation:

Step1: Substitute G with 2

We know the profit \( G \) is 2 (thousand dollars), so substitute \( G = 2 \) into the equation \( G=\sqrt[4]{40a - 319}-1 \). We get \( 2=\sqrt[4]{40a - 319}-1 \).

Step2: Isolate the fourth - root term

Add 1 to both sides of the equation \( 2=\sqrt[4]{40a - 319}-1 \) to isolate the fourth - root term. So \( 2 + 1=\sqrt[4]{40a - 319} \), which simplifies to \( 3=\sqrt[4]{40a - 319} \).

Step3: Eliminate the fourth - root

Raise both sides of the equation \( 3=\sqrt[4]{40a - 319} \) to the fourth power. According to the property \((\sqrt[n]{x})^n=x\) (for \( x\geq0 \) when \( n \) is even), we have \( 3^4=40a - 319 \). Since \( 3^4 = 81 \), the equation becomes \( 81=40a - 319 \).

Step4: Solve for a

Add 319 to both sides of the equation \( 81=40a - 319 \): \( 81+319 = 40a \). Calculate \( 81 + 319=400 \), so \( 400 = 40a \). Then divide both sides by 40: \( a=\frac{400}{40}=10 \).

Answer:

The time when the profit reaches 2 thousand dollars is 10 hours.