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Use the image to answer the question. For the square pyramid, the lateral edge length, e, is 25 in., and the slant height, s, i...Question
Use the image to answer the question. For the square pyramid, the lateral edge length, e , is 25 in., and the slant height, s , is 24 in. Find the height of the pyramid, h . Round the answer to the nearest whole number. Hint: Before finding the length of h , you will first need to find the length of r . (1 point) The height of the square pyramid is in.
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Answered by
GPT-4o mini
To find the height \( h \) of the square pyramid with given lateral edge length \( e = 25 \) inches and slant height \( s = 24 \) inches, we first need to determine the length of \( r \), which is half the length of the base side of the pyramid.
In a square pyramid, the relationship between the slant height \( s \), the height \( h \), and half the base side \( r \) can be expressed using the Pythagorean theorem:
\[
s^2 = h^2 + r^2
\]
From the problem, we also know the relationship between the lateral edge \( e \), the height \( h \), and half the diagonal of the base. The half diagonal \( d \) of the base can be calculated from the side length of the base, which is \( 2r \):
\[
d = \sqrt{(2r)^2 + (2r)^2} = \sqrt{8r^2} = 2\sqrt{2}r
\]
The relationship involving the lateral edge \( e \) can also be expressed using the Pythagorean theorem:
\[
e^2 = h^2 + d^2
\]
Substituting for \( d \):
\[
e^2 = h^2 + (2\sqrt{2}r)^2 = h^2 + 8r^2
\]
Now we have two equations:
1. \( s^2 = h^2 + r^2 \)
2. \( e^2 = h^2 + 8r^2 \)
Now, let's substitute the known values:
1. For \( s = 24 \):
\[
24^2 = h^2 + r^2 \implies 576 = h^2 + r^2 \quad (1)
\]
2. For \( e = 25 \):
\[
25^2 = h^2 + 8r^2 \implies 625 = h^2 + 8r^2 \quad (2)
\]
Now we can solve these two equations simultaneously. From equation (1), we can express \( h^2 \):
\[
h^2 = 576 - r^2
\]
Now, substitute \( h^2 \) into equation (2):
\[
625 = (576 - r^2) + 8r^2
\]
\[
625 = 576 + 7r^2
\]
\[
625 - 576 = 7r^2
\]
\[
49 = 7r^2 \implies r^2 = 7 \implies r = \sqrt{7}
\]
Now substituting \( r^2 = 7 \) back into equation (1) to find \( h^2 \):
\[
h^2 = 576 - 7
\]
\[
h^2 = 569
\]
\[
h = \sqrt{569} \approx 23.8
\]
Rounding to the nearest whole number, the height \( h \) of the square pyramid is:
\[
\boxed{24} \text{ inches}
\]
In a square pyramid, the relationship between the slant height \( s \), the height \( h \), and half the base side \( r \) can be expressed using the Pythagorean theorem:
\[
s^2 = h^2 + r^2
\]
From the problem, we also know the relationship between the lateral edge \( e \), the height \( h \), and half the diagonal of the base. The half diagonal \( d \) of the base can be calculated from the side length of the base, which is \( 2r \):
\[
d = \sqrt{(2r)^2 + (2r)^2} = \sqrt{8r^2} = 2\sqrt{2}r
\]
The relationship involving the lateral edge \( e \) can also be expressed using the Pythagorean theorem:
\[
e^2 = h^2 + d^2
\]
Substituting for \( d \):
\[
e^2 = h^2 + (2\sqrt{2}r)^2 = h^2 + 8r^2
\]
Now we have two equations:
1. \( s^2 = h^2 + r^2 \)
2. \( e^2 = h^2 + 8r^2 \)
Now, let's substitute the known values:
1. For \( s = 24 \):
\[
24^2 = h^2 + r^2 \implies 576 = h^2 + r^2 \quad (1)
\]
2. For \( e = 25 \):
\[
25^2 = h^2 + 8r^2 \implies 625 = h^2 + 8r^2 \quad (2)
\]
Now we can solve these two equations simultaneously. From equation (1), we can express \( h^2 \):
\[
h^2 = 576 - r^2
\]
Now, substitute \( h^2 \) into equation (2):
\[
625 = (576 - r^2) + 8r^2
\]
\[
625 = 576 + 7r^2
\]
\[
625 - 576 = 7r^2
\]
\[
49 = 7r^2 \implies r^2 = 7 \implies r = \sqrt{7}
\]
Now substituting \( r^2 = 7 \) back into equation (1) to find \( h^2 \):
\[
h^2 = 576 - 7
\]
\[
h^2 = 569
\]
\[
h = \sqrt{569} \approx 23.8
\]
Rounding to the nearest whole number, the height \( h \) of the square pyramid is:
\[
\boxed{24} \text{ inches}
\]
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