If the cylinder were dilated using a scale factor of 4, what would be the new surface area? The formula for the surface area of a cylinder is SA=2πrh+2πr2

Question

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(1 point)
Responses

368π cm2
368 pi cm squared

23π cm2
23 pi cm squared

1,472π cm2
1 comma 472 pi cm squared

5,888π cm2
5,888 pi cm squared

Answer 1

To determine the new surface area of the cylinder after a dilation with a scale factor of 4, we need to understand how dilation affects the dimensions of a cylinder.

Let’s denote the original radius and height of the cylinder as \( r \) and \( h \). The surface area \( SA \) of a cylinder is calculated using the formula:

\[ SA = 2\pi rh + 2\pi r^2 \]

When a cylinder is dilated by a scale factor of \( k \), both the radius and height are multiplied by \( k \). With a scale factor of 4, the new dimensions will be:

• New radius: \( 4r \)
• New height: \( 4h \)

Now, we can calculate the new surface area \( SA' \) using the new dimensions:

\[ SA' = 2\pi(4r)(4h) + 2\pi(4r)^2 \]

Calculating each term individually:

1. The lateral surface area: \[ 2\pi(4r)(4h) = 2\pi \cdot 16rh = 32\pi rh \]

2. The area of the top and bottom: \[ 2\pi(4r)^2 = 2\pi \cdot 16r^2 = 32\pi r^2 \]

Now combining these results:

\[ SA' = 32\pi rh + 32\pi r^2 = 32\pi (rh + r^2) \]

Notice that the original surface area \( SA = 2\pi rh + 2\pi r^2 \) can be factored out:

\[ SA' = 16 \times SA \]

Thus, the new surface area is \( 16 \) times the original surface area.

Now, in order to find the final answer, we need the original surface area. The possible answers you've provided are:

• \( 368\pi , \text{cm}^2 \)
• \( 23\pi , \text{cm}^2 \)
• \( 1,472\pi , \text{cm}^2 \)
• \( 5,888\pi , \text{cm}^2 \)

To find the original surface area from which we derive the new surface area, we can go through each option to find which one gives us the correct new surface area when multiplied by 16.

1. If original SA = \( 368\pi \): \[ SA' = 16 \times 368\pi = 5,888\pi , \text{cm}^2 \]

2. If original SA = \( 23\pi \): \[ SA' = 16 \times 23\pi = 368\pi , \text{cm}^2 \]

3. If original SA = \( 1,472\pi \): \[ SA' = 16 \times 1,472\pi = 23,552\pi , \text{cm}^2 \]

4. If original SA = \( 5,888\pi \): \[ SA' = 16 \times 5,888\pi = 94,208\pi , \text{cm}^2 \]

From our calculations, the only option that yields a logical relationship under dilation is:

• The original surface area must be \( 368\pi , \text{cm}^2 \), giving a new surface area of \( 5,888\pi , \text{cm}^2 \).

Therefore, the new surface area of the cylinder after dilation with a scale factor of 4 is:

5,888π cm²