Question

questions
1.a cone has a radius of 3cm and slant height of 5cm, find the total surface area of the cone.
2.the total surface area of a cone is 375 square inches. if its slant height is four times the radius, then what is the base diameter of the cone?
3.what is the total surface area of a cone if its radius = 4cm and height = 3 cm?

Explanation:

Question 1

Step1: Recall the formula for the total surface area (TSA) of a cone.

The formula for the total surface area of a cone is \( TSA=\pi r(r + l) \), where \( r \) is the radius and \( l \) is the slant height.

Step2: Substitute the given values into the formula.

Given that \( r = 3\space cm \) and \( l=5\space cm \).
Substitute these values into the formula:
\( TSA=\pi\times3\times(3 + 5) \)
\(=\pi\times3\times8 \)
\( = 24\pi\space cm^{2}\) (or approximately \( 24\times3.14 = 75.36\space cm^{2}\))

Step1: Recall the formula for the total surface area of a cone.

The formula for the total surface area of a cone is \( TSA=\pi r(r + l) \). Given that \( l = 4r \) and \( TSA=375\space square\space inches \).

Step2: Substitute \( l = 4r \) into the formula.

Substitute \( l = 4r \) into the formula \( TSA=\pi r(r + l) \):
\( 375=\pi r(r + 4r) \)
\( 375=\pi r(5r) \)
\( 375 = 5\pi r^{2}\)

Step3: Solve for \( r \).

First, divide both sides by \( 5\pi \):
\( r^{2}=\frac{375}{5\pi}=\frac{75}{\pi} \)
Then, take the square root of both sides:
\( r=\sqrt{\frac{75}{\pi}}\approx\sqrt{\frac{75}{3.14}}\approx\sqrt{23.885}\approx4.89\space inches \)

Step4: Find the diameter.

The diameter \( d = 2r \), so \( d=2\times4.89\approx9.78\space inches \) (or we can solve it symbolically first)

Let's solve it symbolically:

From \( 375 = 5\pi r^{2}\)

\( r^{2}=\frac{375}{5\pi}=\frac{75}{\pi}\)

\( r=\sqrt{\frac{75}{\pi}}\)

Diameter \( d = 2r=2\sqrt{\frac{75}{\pi}}=\sqrt{\frac{300}{\pi}}\approx\sqrt{\frac{300}{3.14}}\approx\sqrt{95.54}\approx9.77\space inches \)

Step1: Find the slant height \( l \) of the cone.

We know that for a cone, \( l=\sqrt{r^{2}+h^{2}} \), where \( r \) is the radius and \( h \) is the height. Given \( r = 4\space cm \) and \( h = 3\space cm \).
So, \( l=\sqrt{4^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5\space cm \)

Step2: Recall the formula for the total surface area of a cone.

The formula for the total surface area of a cone is \( TSA=\pi r(r + l) \)

Step3: Substitute the values of \( r \) and \( l \) into the formula.

Given \( r = 4\space cm \) and \( l = 5\space cm \)
\( TSA=\pi\times4\times(4 + 5) \)
\(=\pi\times4\times9 \)
\( = 36\pi\space cm^{2}\) (or approximately \( 36\times3.14=113.04\space cm^{2}\))

Answer:

The total surface area of the cone is \( 24\pi\space cm^{2}\) (or approximately \( 75.36\space cm^{2}\))

Question 2