Question

The time \(t\) (in hours) needed to produce \(x\) units of a product is modeled by \(t=p x+s .\) If it takes 265 hours to produce 200 units and 390 hours to produce 300 units, what is the value of \(s ?\) (A) 1.25 (B) 15 (C) 100 (D) 125

Short answer

The correct option is (B) 15

Step-by-step solution

Set up the equations

The first step is to translate the given conditions into mathematical equations using the model \(t = px + s\). We have two conditions: \n1. \(t = 265\) when \(x = 200\), which gives the equation \(265 = 200p + s\)\n2. \(t = 390\) when \(x = 300\), which gives the equation \(390 = 300p + s\)

Solve the system of equations

After setting up the equations, we can write them in matrix form and solve for \(p\) and \(s\):\n\[\begin{cases} 200p + s = 265 \ 300p + s = 390 \end{cases}\]\nBy subtracting the first equation from the second, we find: \(100p = 125\). This gives us \(p = 1.25\). Now, we can replace \(p\) in the first equation with \(1.25\) to obtain: \(265 = 200(1.25) + s\), which upon simplification gives \(s = 15\).

Verify

To ensure the solution is correct, we can substitute \(p = 1.25\) and \(s = 15\) into the second condition and check if the left hand side equals the right hand side. This gives: \(390 = 300(1.25) + 15\), which simplifies to \(390 = 375 + 15\), which is true.