Question

question 40 of 50
a right cylinder with a lateral area of 75 units squared and a height of 6 units has a surface area of 100 units squared (if the sa is rounded to the nearest whole).
true
false

Explanation:

Step1: Recall Lateral and Surface Area Formulas

The lateral (curved) surface area of a right cylinder is \( L = 2\pi rh \), and the total surface area is \( SA = 2\pi rh + 2\pi r^2 = L + 2\pi r^2 \), where \( r \) is the radius and \( h \) is the height. We know \( L = 75 \) and \( h = 6 \). First, find \( r \) from the lateral area formula.

From \( L = 2\pi rh \), solve for \( r \):
\( 75 = 2\pi r(6) \)
\( 75 = 12\pi r \)
\( r = \frac{75}{12\pi} = \frac{25}{4\pi} \approx \frac{25}{12.566} \approx 1.989 \) units.

Step2: Calculate Surface Area

Now, use the surface area formula \( SA = L + 2\pi r^2 \). Substitute \( L = 75 \) and \( r \approx 1.989 \):

First, compute \( 2\pi r^2 \):
\( 2\pi (1.989)^2 \approx 2\pi (3.956) \approx 7.912\pi \approx 24.85 \)

Then, \( SA \approx 75 + 24.85 = 99.85 \approx 100 \) (rounded to the nearest whole number).

Answer:

True