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 . (1 point) Responses 5,888π cm2 5,888 pi cm squared 1,472π cm2 1 comma 472 pi cm squared 23π cm2 23 pi cm squared 368π cm2

Answer 1

To find the new surface area of a cylinder after dilation with a scale factor of 4, we must first understand how scaling affects the surface area.

The surface area \(SA\) of a cylinder is given by the formula: \[ SA = 2\pi rh + 2\pi r^2 \] where \(r\) is the radius and \(h\) is the height of the cylinder.

When the cylinder is dilated by a scale factor of 4, both the radius and the height increase by that factor. Therefore:

• The new radius \(r'\) will be \(4r\).
• The new height \(h'\) will be \(4h\).

Now, we can substitute these new values into the surface area formula:

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

Calculating each term:

1. \(2\pi (4r)(4h) = 2\pi (16rh) = 32\pi rh\)
2. \(2\pi (4r)^2 = 2\pi (16r^2) = 32\pi r^2\)

So, adding these two parts together gives:

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

Notice that the original surface area \(SA = 2\pi rh + 2\pi r^2\) is a factor of \(SA = 2\pi (rh + r^2)\).

Since we've scaled up both dimensions by a factor of 4, we actually know the original SA multiplied by \(16\) (which accounts for the increase in both the length and width contributing to area).

The relationship is: \[ SA' = 16 \times SA \]

Thus, if you know the original surface area, you can simply multiply it by \(16\) to find the new surface area.

For the options given, if we assume the original SA was \(1\pi\), the new SA would be \(16\pi\), but since we don't know that value, we will check the response options given without specific dimensions.

Given that the scaled area is affected substantially with a \(4\) scale factor (leading us to calculate the options based on that principle):

When checking the responses:

1. The surface area will grow significantly, and from the provided options, only \(1,472\pi cm^2\) (which equals \(16\pi\) if \(SA\) were \(92\pi cm^2\)) and \(5,888\pi cm^2\) have the plausibility given high scaling impacts for radius and height.

The final answer can conclude at: If you compute accordingly based on above and units, \(SA' = 5,888\pi cm^2\) with scaling would refer to the previous combined area multiple effects on both \(r\) and \(h\).

Thus, the new surface area \(= 5,888\pi cm^2\).