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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.

3 answers

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 \( 2b \):

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

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

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

  1. Calculate \( 96 - 16 \):

\[ 96 - 16 = 80 \]

  1. Divide by \( h \):

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

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

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

can you use less words and make it less advanced

Sure! Here’s a simpler version.

Part A: Solve for \( p \)

Starting with the equation:

\[ s = ph + 2b \]

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

\[ s - 2b = ph \]

  1. 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 \( 2b \):

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

  1. Substitute in the formula:

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

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

  2. Divide by \( 4 \):

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

Final answer for the perimeter \( p \):

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