The problem asks us to estimate the total number of baseball caps produced by a company over 8 months using the midpoint rule with 8 subintervals, based on the provided graph. The x-axis represents the number of months, and the y-axis represents the number of caps produced.

1. Problem Description

The problem asks us to estimate the total number of baseball caps produced by a company over 8 months using the midpoint rule with 8 subintervals, based on the provided graph. The x-axis represents the number of months, and the y-axis represents the number of caps produced.

2. Solution Steps

The midpoint rule approximates the area under a curve by summing the areas of rectangles whose heights are given by the value of the function at the midpoint of each subinterval. In this case, the "area" represents the total number of caps produced. Since we have 8 subintervals over 8 months, each subinterval has a width of 1 month.

Midpoint Rule:

$Area \approx \Delta x [f(x_1) + f(x_2) + ... + f(x_n)]$

where $\Delta x$ is the width of each subinterval, and $f(x_i)$ is the function value at the midpoint of the $i$-th subinterval. In our case, $\Delta x = 1$.

We need to estimate the function value (number of caps) at the midpoint of each month.

Month 0.5: Approximately 15

Month 1.5: Approximately 35

Month 2.5: Approximately 25

Month 3.5: Approximately 22

Month 4.5: Approximately 15

Month 5.5: Approximately 23

Month 6.5: Approximately 8

Month 7.5: Approximately 5

Summing the values:

$15 + 35 + 25 + 22 + 15 + 23 + 8 + 5 = 148$

Since $\Delta x = 1$, the estimated total number of caps is:

$1 * 148 = 148$

3. Final Answer

148