Math / Geometry

QuestionWhat is the surface area of this cylinder? Use $\pi \approx 3.14$ and round your answer to the nearest hundredth. $\square$ square inches

Studdy Solution

STEP 1

What is this asking? We need to find the total surface area of a cylinder, basically how much wrapping paper we'd need to cover it entirely! Watch out! Don't forget that the surface area includes the top and bottom circles and the curved side!
Also, remember we're given the diameter, not the radius, so we'll need to convert that first.

STEP 2

1. Find the radius
2. Calculate the area of the circular bases
3. Calculate the area of the curved side
4. Calculate the total surface area

STEP 3

Alright, let's start with finding the radius!
We know the diameter is $16.4$ inches.
The radius is half the diameter, so we divide the diameter by $2$ .

STEP 4

$\text{Radius} = \frac{\text{Diameter}}{2} = \frac{16.4}{2} = 8.2$ So, our radius is $8.2$ inches!

STEP 5

Now, let's find the area of one of the circular bases.
Remember, the formula for the area of a circle is $\pi \cdot r^2$ , where $r$ is the radius.
We're using $\pi \approx 3.14$ and our radius is $8.2$ inches.

STEP 6

$\text{Area of one base} = \pi \cdot r^2 \approx 3.14 \cdot (8.2)^2 \approx 3.14 \cdot 67.24 \approx 211.1336$ So, the area of one circular base is approximately $211.1336$ square inches.

STEP 7

Since there are two bases (top and bottom), we multiply this by $2$ : $\text{Area of both bases} \approx 2 \cdot 211.1336 \approx 422.2672$ The combined area of both bases is approximately $422.2672$ square inches.

STEP 8

Imagine unrolling the curved side of the cylinder—it becomes a rectangle!
The height of this rectangle is the same as the cylinder's height, which is $14.9$ inches.
The width of the rectangle is the circumference of the circular base, which is $2 \cdot \pi \cdot r$ .

STEP 9

$\text{Circumference} = 2 \cdot \pi \cdot r \approx 2 \cdot 3.14 \cdot 8.2 \approx 51.496$ The circumference is approximately $51.496$ inches.

STEP 10

Now, let's find the area of the curved side (the rectangle): $\text{Area of curved side} = \text{Circumference} \cdot \text{Height} \approx 51.496 \cdot 14.9 \approx 767.2904$ So, the area of the curved side is approximately $767.2904$ square inches.

STEP 11

Finally, we add the areas of the two bases and the curved side to get the total surface area: $\text{Total Surface Area} \approx 422.2672 + 767.2904 \approx 1189.5576$

STEP 12

Rounding to the nearest hundredth gives us $1189.56$ .

STEP 13

The surface area of the cylinder is approximately $1189.56$ square inches.

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