Find the surface area of the fabric required to make a lamp shade in the shape of a frustrum whose top and bottom diameter are 20cm and 30cm respectively and a height of 12cm.Give your answer to 2 decimals.
Consider a complete cone of height 36. The lamp shade is the bottom 1/3 of that cone. (draw a side view and use similar triangles)
cone has area A = 2πrs = 2π*15*39
The lampshade has area a = A - 2π*10*26
or you can figure a general formula using h,R,r as given in the problem.
Well, I hope the lamp shade doesn't become frustrated while we calculate its surface area!
To find the surface area of a frustrum, we need to find the area of the top circle, the area of the bottom circle, and the area of the lateral surface. Then we can add them all together.
First, let's calculate the radii of the top and bottom circles. The radius of the top circle is half of the top diameter, which is 20cm/2 = 10cm. Similarly, the radius of the bottom circle is half of the bottom diameter, which is 30cm/2 = 15cm.
Next, let's calculate the area of the top and bottom circles using the formula: area = π * radius^2. For the top circle, the area is π * (10cm)^2 = 100π cm^2. For the bottom circle, the area is π * (15cm)^2 = 225π cm^2.
Now, let's calculate the lateral surface area using the formula: area = π * (top radius + bottom radius) * slant height. The slant height can be found by using the Pythagorean theorem. The slant height is √(12cm^2 + (15cm - 10cm)^2) = √(144cm^2 + 25cm^2) = √169cm^2 = 13cm. So the lateral surface area is π * (10cm + 15cm) * 13cm = 650π cm^2.
Finally, let's add up the area of the top circle, the area of the bottom circle, and the lateral surface area. That gives us 100π cm^2 + 225π cm^2 + 650π cm^2 = 975π cm^2.
And since you want the answer to 2 decimal places, let's calculate that. According to my calculations, the surface area of the fabric required for the lamp shade is approximately 3063.58 cm².
So, please make sure to get at least 3064 cm² of fabric to avoid any frustrated lamp shade mishaps!
To find the surface area of the fabric required to make a lamp shade in the shape of a frustum, we need to calculate the area of the top circle, the area of the bottom circle, and the lateral surface area.
1. Calculate the area of the top circle:
The radius of the top circle is half of the top diameter, which is 20cm / 2 = 10cm.
The area of a circle is given by the formula A = πr^2, where π is approximately 3.14.
So, the area of the top circle is A1 = 3.14 * (10cm)^2.
2. Calculate the area of the bottom circle:
The radius of the bottom circle is half of the bottom diameter, which is 30cm / 2 = 15cm.
The area of the bottom circle is A2 = 3.14 * (15cm)^2.
3. Calculate the lateral surface area:
The lateral surface area of a frustum is given as A = π(R+r)*l, where R is the radius of the bottom circle, r is the radius of the top circle, and l is the slant height of the frustum.
To calculate the slant height, we can use the Pythagorean theorem:
l^2 = (h^2) + (R - r)^2,
where h is the height of the frustum.
Plugging in the given values, we have l^2 = (12cm)^2 + (15cm - 10cm)^2.
Simplifying, we get l^2 = 144cm^2 + 25cm^2.
Taking the square root of both sides, we find l ≈ √(169cm^2) ≈ 13cm.
Now, we can calculate the lateral surface area:
A3 = 3.14 * (15cm + 10cm) * 13cm.
4. Calculate the total surface area:
To find the total surface area, we just need to add the areas of the top and bottom circles, and the lateral surface area:
Total surface area = A1 + A2 + A3.
Plugging in the values we calculated:
Total surface area ≈ A1 + A2 + A3
≈ 3.14 * (10cm)^2 + 3.14 * (15cm)^2 + 3.14 * (15cm + 10cm) * 13cm
≈ 314cm^2 + 706.5cm^2 + 3.14 * 25cm * 13cm
≈ 314cm^2 + 706.5cm^2 + 1021.25cm^2
≈ 2041.75cm^2.
Therefore, the surface area of the fabric required to make the lamp shade is approximately 2041.75 square centimeters.
To find the surface area of the fabric required to make a lamp shade in the shape of a frustum, we need to calculate the lateral surface area and the area of the top and bottom circles. Then, we add these two areas together to get the total surface area.
The lateral surface area of a frustum can be calculated using the formula:
A = π(R1 + R2) l
where R1 and R2 are the radii of the top and bottom circles of the frustum, and l is the slant height of the frustum.
First, let's calculate the slant height. We can use the Pythagorean theorem to find the slant height (l):
l = sqrt(h^2 + (R2 - R1)^2)
Given:
Top diameter (R1) = 20 cm
Bottom diameter (R2) = 30 cm
Height (h) = 12 cm
To convert diameters to radii, we divide by 2:
R1 = 20 cm / 2 = 10 cm
R2 = 30 cm / 2 = 15 cm
Plugging in these values, we can find the slant height (l):
l = sqrt(12^2 + (15 - 10)^2)
= sqrt(144 + 25)
= sqrt(169)
= 13 cm
Now, let's calculate the lateral surface area (A) using the formula:
A = π(R1 + R2) l
= π(10 + 15) × 13
= π × 25 × 13
= 325π cm^2
Next, let's calculate the area of the top and bottom circles. The formula for the area of a circle is:
A_circle = π × (radius)^2
The radius of the top circle is R1 = 10 cm, so the area is:
A_top = π × (10)^2
= 100π cm^2
The radius of the bottom circle is R2 = 15 cm, so the area is:
A_bottom = π × (15)^2
= 225π cm^2
To find the total surface area, we add the lateral surface area and the areas of the top and bottom circles:
Total surface area = A + A_top + A_bottom
= 325π + 100π + 225π
= 650π cm^2
To get the answer rounded to 2 decimal places, we can use the value of π as approximately 3.14:
Total surface area ≈ 650 × 3.14
≈ 2041.00 cm^2
Therefore, the surface area of the fabric required to make the lamp shade in the shape of a frustum is approximately 2041.00 cm².