Question

a right cone has a base radius of 4R and a height of 3R.what is the ratio of total surface area of just the base​

Answer 1

Answer:

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Answer 2

Answer:

The ratio of total surface area of just the base​ is 9 : 4

Step-by-step explanation:

Given: A right cone has a base radius of 4R and a height of 3R

To find: The total surface of the cone and the area of the cone.

Solution:

The area of the base of the cone is equal to the area of the circle

Area of the circle A =  $\pi r^{2}$

Where r is the length of the radius.

In this cone the radius is equal to 4R so we have to replace r with R

A = $\pi (4R)^{2}$

A = $16\pi R^{2}$

To calculate the total area of the cone, we have to know the area of the base and the lateral surface area of the cone

The Lateral surface area of the cone(LA)

LA = $\frac{1}{2} (2\pi r)(l)$

Where r is the radius

l is the slant height

We know that r = 4R

now we have to find slant height

Slant height (l) is  the distance from the edge of the base of the cone to the tip.

Use Pythagoras theorem

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs, which in this case are 4R and 3R. (which is the slant height).

$(4R)^{2} + (3R)^{2} = l^{2}$

$16R^{2} + 9R^{2} = l^{2}$

$25R^{2} = l^{2}$

l = 5R

LA = $\frac{1}{2} (2\pi r)(l)$

LA = $\frac{1}{2} 2\pi 4R(5R)$

LA = $20\pi R^{2}$

Total surface area = $20\pi R^{2} + 16\pi R^{2}$

TA = $36\pi R^{2}$

We have to find the ratio of the total surface area to the area of the base.

⇒ $\frac{36\pi R^{2}}{16\pi R^{2} }$

⇒ $\frac{36}{16}$

⇒$\frac{9}{4}$

The ratio is 9 : 4

Final answer:

The ratio of total surface area of just the base​ is 9 : 4

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