The surface area of a rectangular prism is 220 cm². Two of the dimensions are 5 cm and 10 cm. Find the measure of the other dimension.

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Understand the Problem

The question is asking for the measure of the third dimension of a rectangular prism, given the surface area and two of its dimensions. We will set up the surface area formula for a rectangular prism and solve for the unknown dimension.

Answer

The measure of the other dimension is $4 \ \text{cm}$.
Answer for screen readers

The measure of the other dimension is $4 \ \text{cm}$.

Steps to Solve

  1. Identify the surface area formula for a rectangular prism

The formula for the surface area $S$ of a rectangular prism with dimensions length $l$, width $w$, and height $h$ is given by:

$$ S = 2(lw + lh + wh) $$

  1. Substitute known values into the formula

We know:

  • Surface area ($S$) = 220 cm²
  • Length ($l$) = 5 cm
  • Width ($w$) = 10 cm

Substituting these values into the surface area formula gives:

$$ 220 = 2(5 \times 10 + 5 \times h + 10 \times h) $$

  1. Simplify the equation

Calculating the area term yields:

$$ 220 = 2(50 + 5h + 10h) $$

So, we simplify it further:

$$ 220 = 2(50 + 15h) $$

  1. Remove the factor of 2

Divide both sides by 2 to simplify the equation:

$$ 110 = 50 + 15h $$

  1. Isolate $h$

To find $h$, subtract 50 from both sides:

$$ 110 - 50 = 15h $$

This results in:

$$ 60 = 15h $$

  1. Solve for $h$

Now, divide by 15:

$$ h = \frac{60}{15} = 4 $$

The measure of the other dimension is $4 \ \text{cm}$.

More Information

The problem utilizes the concept of surface area in three-dimensional geometry, specifically for rectangular prisms. Understanding how to manipulate the surface area formula can help solve for unknown dimensions efficiently.

Tips

One common mistake is neglecting to correctly apply the surface area formula, forgetting that each rectangular face contributes to the total area. Always ensure all areas are accounted in the formula.

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