Sure, let's go through the problem step-by-step to find the total surface area of the cone.
1. Given Information:
- The radius ([tex]\( r \)[/tex]) of the cone is 7 cm.
- The sum of the radius and the slant height ([tex]\( r + l \)[/tex]) is 28 cm.
2. Finding the Slant Height:
- We know that [tex]\( r + l = 28 \)[/tex] cm.
- Given [tex]\( r = 7 \)[/tex] cm, we can find the slant height ([tex]\( l \)[/tex]) by subtracting the radius from the sum:
[tex]\[
l = 28 - 7 = 21 \text{ cm}
\][/tex]
3. Formula for Total Surface Area of a Cone:
- The total surface area [tex]\( A \)[/tex] of a cone is given by:
[tex]\[
A = \pi r (r + l)
\][/tex]
- Here, [tex]\( r = 7 \)[/tex] cm and [tex]\( l = 21 \)[/tex] cm.
4. Substituting the Values:
- Plugging in the values into the formula:
[tex]\[
A = \pi \times 7 \times (7 + 21)
\][/tex]
- Simplify inside the parentheses:
[tex]\[
A = \pi \times 7 \times 28
\][/tex]
5. Calculating the Surface Area:
- We obtain:
[tex]\[
A \approx 3.14159 \times 7 \times 28
\][/tex]
- Multiplying these together, we get:
[tex]\[
A \approx 615.7521601035994 \text{ cm}^2
\][/tex]
6. Selecting the Closest Answer:
- Among the given options, the closest value to [tex]\( 615.7521601035994 \)[/tex] cm² is:
[tex]\[
\boxed{616 \text{ cm}^2}
\][/tex]
So, the total surface area of the cone is [tex]\( \boxed{616 \text{ cm}^2} \)[/tex].
