Question

The surface area of a prism is modeled by the equation
s= ph + 2b, where
s is the surface area,
p is the perimeter of the base,
h is height of the prism, and
b is the area of the base.

A. Solve the equation s= ph + 2b
for the perimeter p
of the base.
B. A rectangular prism has a height of 4 meters, a surface area of 96 square meters, and a base area of 8 square meters. Calculate the perimeter of the base of the prism.

In your final answers, include all of your calculations.
Answer
can you use less words and make it less advanced
Answered by GPT-4o mini
To address the problem, we'll follow the two parts (A and B) as directed.

### Part A: Solve the equation \( s = ph + 2b \) for \( p \)

We start with the given equation:

\[
s = ph + 2b
\]

To isolate \( p \), we first subtract \( 2b \) from both sides:

\[
s - 2b = ph
\]

Next, we divide both sides by \( h \) to solve for \( p \):

\[
p = \frac{s - 2b}{h}
\]

So the formula for the perimeter \( p \) of the base is:

\[
\boxed{p = \frac{s - 2b}{h}}
\]

---

### Part B: Calculate the perimeter of the base of a rectangular prism

Given that:

- Height \( h = 4 \) meters
- Surface area \( s = 96 \) square meters
- Base area \( b = 8 \) square meters

We will use the formula derived in part A:

\[
p = \frac{s - 2b}{h}
\]

Substituting the provided values into the equation:

1. Calculate \( 2b \):

\[
2b = 2 \times 8 = 16
\]

2. Substitute \( s = 96 \), \( 2b = 16 \), and \( h = 4 \) into the formula:

\[
p = \frac{96 - 16}{4}
\]

3. Calculate \( 96 - 16 \):

\[
96 - 16 = 80
\]

4. Divide by \( h \):

\[
p = \frac{80}{4} = 20
\]

Therefore, the perimeter of the base of the prism is:

\[
\boxed{20} \text{ meters}
\]
Answered by GPT-4o mini
Sure! Here’s a simpler version.

### Part A: Solve for \( p \)

Starting with the equation:

\[
s = ph + 2b
\]

1. Subtract \( 2b \) from both sides:

\[
s - 2b = ph
\]

2. Divide by \( h \):

\[
p = \frac{s - 2b}{h}
\]

**Final answer for \( p \):**

\[
\boxed{p = \frac{s - 2b}{h}}
\]

---

### Part B: Calculate \( p \)

Given:

- Height \( h = 4 \) m
- Surface area \( s = 96 \) m²
- Base area \( b = 8 \) m²

1. Find \( 2b \):

\[
2b = 2 \times 8 = 16
\]

2. Substitute in the formula:

\[
p = \frac{96 - 16}{4}
\]

3. Calculate \( 96 - 16 = 80 \).

4. Divide by \( 4 \):

\[
p = \frac{80}{4} = 20
\]

**Final answer for the perimeter \( p \):**

\[
\boxed{20 \text{ meters}}
\]