It seems you're referring to the Trapezoidal Rule formula for estimating the area under a curve, but the formula you provided is a bit unclear. Let's go over the standard Trapezoidal Rule formula first, and then we can interpret your expression.

Trapezoidal Rule Formula

The Trapezoidal Rule is given by:

$\text{Area} \approx \frac{h}{2} \times \left[ f(x_0) + 2 \times \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$

Where:

• $h$ is the width of each interval.
• $f(x_0)$ and $f(x_n)$ are the function values at the endpoints.
• $\sum_{i=1}^{n-1} f(x_i)$ is the sum of the function values at the interior points.

Interpreting Your Expression

Given your expression:

$\text{Area} = \frac{32}{2(18 + d_\text{last})}$

It seems like you're using a simplified version, where:

• The numerator $32$ could represent some combination of height or sum of areas.
• The denominator $2(18 + d_\text{last})$ might represent the total width including some last distance $d_\text{last}$.

Solving for $d_\text{last}$

If you're asked to solve for $d_\text{last}$, let's rewrite the equation to isolate it:

$\text{Area} = \frac{32}{2(18 + d_\text{last})}$

Given that $\text{Area} = A$, then:

$A = \frac{32}{2(18 + d_\text{last})}$

Multiply both sides by $2(18 + d_\text{last})$ to get:

$2A(18 + d_\text{last}) = 32$

Now, divide both sides by $2A$:

$18 + d_\text{last} = \frac{32}{2A}$

Finally, subtract 18 from both sides to solve for $d_\text{last}$:

$d_\text{last} = \frac{32}{2A} - 18$

Would you like any further details or have any questions about this solution? Here are some related questions:

1. What is the trapezoidal rule, and when is it used?
2. How does changing the function $f(x)$ affect the area calculation?
3. What happens to the area if the last distance $d_\text{last}$ is doubled?
4. Can the trapezoidal rule be applied to non-linear functions?
5. How does the number of trapezoids affect the accuracy of the rule?
6. What is the difference between the trapezoidal rule and Simpson's rule?
7. How does changing the interval width $h$ affect the final area?
8. How would you estimate the area under a curve that is not easily integrable?

Tip: When using the Trapezoidal Rule, more intervals generally lead to a more accurate approximation of the area.