a right squarepyramid has a base length of 4cm and heeight of 10cm.how many times greater is the surface area of a right square pyramid with the same height if the base length is doubled?
To find out how many times greater the surface area of a right square pyramid will be if its base length is doubled, we need to calculate the surface areas of both pyramids.
The surface area of a right square pyramid can be calculated using the formula:
Surface Area = Base Area + (0.5 × Perimeter of Base × Slant Height)
Let's start with the original pyramid:
Base Length (l) = 4 cm
Height (h) = 10 cm
1. Calculate the original pyramid's base area:
Base Area (BA) = l^2
BA = 4^2
BA = 16 cm^2
2. Calculate the original pyramid's slant height:
Slant Height (s) can be found using the Pythagorean Theorem. Since it is a right square pyramid, the slant height is the hypotenuse of a right triangle with the base length (l) and half the height (h/2) as its legs:
s^2 = l^2 + (h/2)^2
s^2 = 4^2 + (10/2)^2
s^2 = 16 + 25
s^2 = 41
s ≈ 6.40 cm (rounded to two decimal places)
3. Calculate the original pyramid's surface area:
Surface Area (SA) = BA + (0.5 × Perimeter of Base × s)
The base of a square pyramid is a square, so its perimeter is 4 times the length of one side.
Perimeter of Base (P) = 4 × l
P = 4 × 4
P = 16 cm
SA = 16 + (0.5 × 16 × 6.40)
SA = 16 + (0.5 × 102.4)
SA = 16 + 51.2
SA = 67.2 cm^2
Now let's calculate the surface area of the pyramid with the doubled base length:
New Base Length = 2 × 4 = 8 cm
1. Calculate the new pyramid's base area:
BA = (New Base Length)^2
BA = 8^2
BA = 64 cm^2
2. Calculate the new pyramid's slant height:
Using the original height:
s^2 = l^2 + (h/2)^2
s^2 = 8^2 + (10/2)^2
s^2 = 64 + 25
s^2 = 89
s ≈ 9.43 cm (rounded to two decimal places)
3. Calculate the new pyramid's surface area:
SA = BA + (0.5 × Perimeter of Base × s)
P = 4 × New Base Length
P = 4 × 8
P = 32 cm
SA = 64 + (0.5 × 32 × 9.43)
SA = 64 + (0.5 × 301.76)
SA = 64 + 150.88
SA = 214.88 cm^2
Now to find how many times greater the surface area is, divide the surface area of the new pyramid by the surface area of the original pyramid:
Times Greater = (New Surface Area) / (Original Surface Area)
Times Greater = 214.88 / 67.2
Times Greater ≈ 3.2
Therefore, the surface area of a right square pyramid with double the base length is approximately 3.2 times greater than the original pyramid.