Question

The figure below shows a solid con if the total surface area is 1,496 CM determine the slant in height of the cone if the diameter is 28

Answer 1

To determine the slant height of the cone, we first need to establish the relationship between the surface area, radius, and slant height.

The total surface area \( A \) of a cone is given by the formula:

\[
A = \pi r l + \pi r^2
\]

where:
- \( r \) is the radius of the base,
- \( l \) is the slant height,
- \( \pi r l \) is the lateral surface area,
- \( \pi r^2 \) is the area of the base.

Given:

- Total surface area \( A = 1,496 \, \text{cm}^2 \)
- Diameter \( d = 28 \, \text{cm} \)

First, let's find the radius \( r \):

\[
r = \frac{d}{2} = \frac{28}{2} = 14 \, \text{cm}
\]

Now we substitute \( r \) into the surface area formula:

\[
1,496 = \pi (14) l + \pi (14)^2
\]

Next, we calculate \( \pi (14)^2 \):

\[
\pi (14)^2 = \pi \times 196 = 196\pi
\]

Substituting this back into the surface area equation gives us:

\[
1,496 = 14\pi l + 196\pi
\]

Next, we can factor out \( \pi \):

\[
1,496 = \pi (14l + 196)
\]

Now, we will divide both sides by \( \pi \):

\[
\frac{1,496}{\pi} = 14l + 196
\]

Substituting \( \pi \approx 3.14 \):

\[
\frac{1,496}{3.14} \approx 476.07
\]

Now, we can substitute back into the formula:

\[
476.07 = 14l + 196
\]

To isolate \( 14l \), we subtract \( 196 \):

\[
476.07 - 196 = 14l
\]

Calculating the left-hand side:

\[
280.07 = 14l
\]

Now, we divide by 14 to find \( l \):

\[
l = \frac{280.07}{14} \approx 20.00 \, \text{cm}
\]

Therefore, the slant height of the cone is approximately \( 20.00 \, \text{cm} \).