# A cone has a height of #12 cm# and its base has a radius of #6 cm#. If the cone is horizontally cut into two segments #9 cm# from the base, what would the surface area of the bottom segment be?

Now, surface area of bottom segment of original cone

By signing up, you agree to our Terms of Service and Privacy Policy

To find the surface area of the bottom segment of the cone after it's horizontally cut 9 cm from the base, first, calculate the slant height of the smaller cone formed. Use the Pythagorean theorem: slant height = sqrt(height^2 + radius^2). Then, calculate the curved surface area of the smaller cone using the formula for the lateral surface area of a cone: π * radius * slant height. Finally, add the area of the circular base.

Surface Area = π * (radius) * (slant height) + π * (radius)^2

By signing up, you agree to our Terms of Service and Privacy Policy

To find the surface area of the bottom segment of the cone after it has been horizontally cut 9 cm from the base, we first need to determine the dimensions of the bottom segment.

The height of the bottom segment can be calculated as the total height of the cone (12 cm) minus the distance from the base where the cut was made (9 cm), which gives us a height of 3 cm for the bottom segment.

Next, we can calculate the radius of the bottom segment using similar triangles. Since the ratio of corresponding sides in similar triangles is constant, we have:

[ \frac{\text{radius of bottom segment}}{\text{radius of cone}} = \frac{\text{height of bottom segment}}{\text{height of cone}} ]

Substitute the values:

[ \frac{\text{radius of bottom segment}}{6 \text{ cm}} = \frac{3 \text{ cm}}{12 \text{ cm}} ]

Solving for the radius of the bottom segment:

[ \text{radius of bottom segment} = \frac{6 \text{ cm} \times 3 \text{ cm}}{12 \text{ cm}} = 1.5 \text{ cm} ]

Now that we have the radius and height of the bottom segment, we can calculate its surface area. The surface area of a cone is given by the formula:

[ \text{Surface Area} = \pi r \left( r + \sqrt{h^2 + r^2} \right) ]

Substitute the values for the radius (( r = 1.5 \text{ cm} )) and height (( h = 3 \text{ cm} )):

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + \sqrt{(3 \text{ cm})^2 + (1.5 \text{ cm})^2} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + \sqrt{9 \text{ cm}^2 + 2.25 \text{ cm}^2} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + \sqrt{11.25 \text{ cm}^2} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \left( 1.5 \text{ cm} + 3 \text{ cm} \right) ]

[ \text{Surface Area} = \pi \times 1.5 \text{ cm} \times 4.5 \text{ cm} ]

[ \text{Surface Area} = 6.75 \pi \text{ cm}^2 ]

So, the surface area of the bottom segment of the cone is ( 6.75 \pi \text{ cm}^2 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A triangle has two corners with angles of # ( pi ) / 3 # and # ( pi )/ 6 #. If one side of the triangle has a length of #1 #, what is the largest possible area of the triangle?
- A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #18 # and the height of the cylinder is #1 #. If the volume of the solid is #21 pi#, what is the area of the base of the cylinder?
- An ellipsoid has radii with lengths of #6 #, #9 #, and #4 #. A portion the size of a hemisphere with a radius of #5 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?
- A rectangle with sides 7 and #x+2# has a perimeter of 40 units. What is the value of #x#?
- A chord with a length of #1 # runs from #pi/4 # to #pi/2 # radians on a circle. What is the area of the circle?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7