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You remove the lid of the can. What is the percent of change in the surface area of the can?
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Overview

Listen $\newline$ You remove the lid of the can. What is the percent of change in the surface area of the can? $\newline$ about $\square$ $\%$ decrease $\newline$ Previous $\newline$ $1$ $\newline$ $2$ $\newline$ $3$ $\newline$ $4$ $\newline$ $5$ $\newline$ $6$ $\newline$ $7$ $\newline$ $8$ $\newline$ Overview

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Percent of change: word problems

Full solution

Q. Listen

\newline

You remove the lid of the can. What is the percent of change in the surface area of the can?

\newline

about

\square

\%

decrease

\newline

Previous

\newline

1

\newline

2

\newline

3

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4

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5

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6

\newline

7

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8

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Overview

Consider Surface Area: First, let's consider the surface area of a can with a lid. The surface area of a can is made up of the area of the circular top and bottom (lids) and the area of the rectangular wrap around the side.
Circle and Rectangle Formulas: The formula for the area of a circle is $A = \pi r^2$ , where $r$ is the radius of the circle. Since there are $2$ lids, we would normally multiply this by $2$ .
Total Surface Area with Lid: The formula for the area of the rectangle that wraps around the side is $A = 2\pi rh$ , where $h$ is the height of the can.
New Surface Area without Lid: So, the total surface area with the lid is $A_{\text{total with lid}} = 2(\pi r^2) + 2\pi rh$ .
Find Percent Change: After removing the lid, we lose the area of one circle, so the new surface area is $A_{\text{total without lid}} = \pi r^2 + 2\pi rh$ .
Substitute Values: Now we need to find the percent change. The percent change formula is $(\text{New Value} - \text{Old Value}) / \text{Old Value} \times 100\%$ .
Simplify Equation: Substitute the surface area values into the percent change formula: Percent Change = $\left(\frac{(\pi r^2 + 2\pi rh) - (2\pi r^2 + 2\pi rh)}{2\pi r^2 + 2\pi rh}\right) \times 100\%$ .
Correct Mistake: Simplify the equation: Percent Change = $\frac{(\pi r^2 + 2\pi rh - 2\pi r^2 - 2\pi rh)}{(2\pi r^2 + 2\pi rh)} \times 100\%$ .
Correct Mistake: Simplify the equation: Percent Change = $(\pi r^2 + 2\pi rh - 2\pi r^2 - 2\pi rh) / (2\pi r^2 + 2\pi rh) \times 100\%$ . This simplifies to Percent Change = $(-\pi r^2) / (2\pi r^2 + 2\pi rh) \times 100\%$ .
Correct Mistake: Simplify the equation: Percent Change = $(\pi r^2 + 2\pi rh - 2\pi r^2 - 2\pi rh) / (2\pi r^2 + 2\pi rh) \times 100\%$ . This simplifies to Percent Change = $(-\pi r^2) / (2\pi r^2 + 2\pi rh) \times 100\%$ . We can cancel out $\pi r^2$ in the numerator and denominator: Percent Change = $-1 / (2 + 2h/r) \times 100\%$ .
Correct Mistake: Simplify the equation: Percent Change = $(\pi r^2 + 2\pi rh - 2\pi r^2 - 2\pi rh) / (2\pi r^2 + 2\pi rh) \times 100\%$ . This simplifies to Percent Change = $(-\pi r^2) / (2\pi r^2 + 2\pi rh) \times 100\%$ . We can cancel out $\pi r^2$ in the numerator and denominator: Percent Change = $-1 / (2 + 2h/r) \times 100\%$ . Oops, looks like we made a mistake in the previous step. We can't cancel out $\pi r^2$ in the numerator and denominator because the denominator has an additional term $2\pi rh$ . We need to correct this.

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