Look at the square pyramid in the image shown. With the given measurements, determine the length of the base of the square pyramid.

Question

a2+b2=c2

1 leg is 20mi
Height slant is 25mi

Answer 1

To find the length of the base of the square pyramid, we can use the Pythagorean theorem based on the right triangle formed by half of the base, the height of the pyramid, and the slant height.

Let's define the variables:

• Let \( x \) be the length of the base of the pyramid.
• The height (the vertical leg) of the pyramid is given as \( 20 \text{ mi} \).
• The slant height (the hypotenuse) is given as \( 25 \text{ mi} \).

Since the right triangle formed will have:

• One leg as half of the base: \( \frac{x}{2} \)
• The other leg as the height: \( 20 \text{ mi} \)
• The hypotenuse as the slant height: \( 25 \text{ mi} \)

Using the Pythagorean theorem:

\[ \left(\frac{x}{2}\right)^{2} + (20)^{2} = (25)^{2} \]

Calculating each term:

1. \( (20)^{2} = 400 \)
2. \( (25)^{2} = 625 \)

Now we can substitute back into the equation:

\[ \left(\frac{x}{2}\right)^{2} + 400 = 625 \]

To isolate \( \left(\frac{x}{2}\right)^{2} \), subtract \( 400 \) from both sides:

\[ \left(\frac{x}{2}\right)^{2} = 625 - 400 \] \[ \left(\frac{x}{2}\right)^{2} = 225 \]

Now take the square root of both sides:

\[ \frac{x}{2} = 15 \]

To find \( x \), multiply both sides by \( 2 \):

\[ x = 30 \]

Therefore, the length of the base of the square pyramid is \( \boxed{30} \text{ mi} \).

Answer 2

How many miles long is the base of the square pyramid?

height slant is 73mi
height is 55mi

Answer 3

To find the length of the base of the square pyramid, we can again use the Pythagorean theorem. In this case, we have the following:

• Height of the pyramid (vertical leg): \( h = 55 \text{ mi} \)
• Slant height (hypotenuse): \( l = 73 \text{ mi} \)
• Half of the base length: \( \frac{x}{2} \)

Using the Pythagorean theorem, we have:

\[ \left(\frac{x}{2}\right)^{2} + h^{2} = l^{2} \]

Substituting the values:

\[ \left(\frac{x}{2}\right)^{2} + (55)^{2} = (73)^{2} \]

Calculating \( (55)^{2} \) and \( (73)^{2} \):

\[ (55)^{2} = 3025 \] \[ (73)^{2} = 5329 \]

Now substituting these into the equation:

\[ \left(\frac{x}{2}\right)^{2} + 3025 = 5329 \]

Subtract \( 3025 \) from both sides:

\[ \left(\frac{x}{2}\right)^{2} = 5329 - 3025 \] \[ \left(\frac{x}{2}\right)^{2} = 2304 \]

Taking the square root of both sides:

\[ \frac{x}{2} = \sqrt{2304} \] \[ \frac{x}{2} = 48 \]

To find \( x \), multiply both sides by \( 2 \):

\[ x = 2 \times 48 = 96 \]

Therefore, the length of the base of the square pyramid is \( \boxed{96} \) miles.

Answer 4

How many miles long is the slant height of the square pyramid?

height is 55mi
leg 1 is 48mi

Answer 5

To find the slant height of the square pyramid, we can again use the Pythagorean theorem. In this case, we know the following:

• Height (vertical leg): \( h = 55 \text{ mi} \)
• One leg of the right triangle (half of the base length): \( \frac{x}{2} = 48 \text{ mi} \) (this is one leg in the triangle that consists of the height and the slant height).

Using the Pythagorean theorem, we have:

\[ l^2 = h^2 + \left(\frac{x}{2}\right)^2 \]

Where:

• \( l \) is the slant height we want to find,
• \( h = 55 \text{ mi} \),
• \( \frac{x}{2} = 48 \text{ mi} \).

Now substituting the known values:

\[ l^2 = (55)^2 + (48)^2 \]

Calculating each term:

1. \( (55)^2 = 3025 \)
2. \( (48)^2 = 2304 \)

Now adding these together:

\[ l^2 = 3025 + 2304 = 5329 \]

Now take the square root to find the slant height \( l \):

\[ l = \sqrt{5329} = 73 \]

Therefore, the slant height of the square pyramid is \( \boxed{73} \) miles.

Answer 6

How many feet long is the radius of the cone?

height is 36ft
height slant is 85ft

Answer 7

To find the radius of the cone, we can use the Pythagorean theorem. In a cone, the slant height, height, and radius form a right triangle. Here are the values we know:

• Height of the cone (vertical leg): \( h = 36 \text{ ft} \)
• Slant height (hypotenuse): \( l = 85 \text{ ft} \)
• Radius of the base of the cone (which we need to find): \( r \)

According to the Pythagorean theorem:

\[ l^2 = r^2 + h^2 \]

Substituting the known values into the equation:

\[ (85)^2 = r^2 + (36)^2 \]

Calculating \( (85)^2 \) and \( (36)^2 \):

\[ (85)^2 = 7225 \]

\[ (36)^2 = 1296 \]

Now substituting these values back into the equation:

\[ 7225 = r^2 + 1296 \]

Next, subtract \( 1296 \) from both sides:

\[ 7225 - 1296 = r^2 \] \[ 5929 = r^2 \]

Now take the square root of both sides to find the radius \( r \):

\[ r = \sqrt{5929} = 77 \]

Therefore, the radius of the cone is \( \boxed{77} \) feet.

Answer 8

How many feet long is the slant height of the cone?

height is 80ft
leg 1 is 39ft

Answer 9

To find the slant height of the cone, we can again use the Pythagorean theorem. In this case, we know the following:

• Height of the cone (vertical leg): \( h = 80 \text{ ft} \)
• One leg of the right triangle, which represents the radius of the base of the cone: \( r = 39 \text{ ft} \)

According to the Pythagorean theorem:

\[ l^2 = h^2 + r^2 \]

Where:

• \( l \) is the slant height we want to find,
• \( h = 80 \text{ ft} \),
• \( r = 39 \text{ ft} \).

Now substituting the known values into the equation:

\[ l^2 = (80)^2 + (39)^2 \]

Calculating each squared term:

\[ (80)^2 = 6400 \]

\[ (39)^2 = 1521 \]

Now adding these together:

\[ l^2 = 6400 + 1521 = 7921 \]

Now take the square root to find the slant height \( l \):

\[ l = \sqrt{7921} = 89 \]

Therefore, the slant height of the cone is \( \boxed{89} \) feet.

Answer 10

How many miles long is the height of the square pyramid?

height slant is 29mi
leg 1 is 20mi

Answer 11

To find the height of the square pyramid, we can use the Pythagorean theorem. We have:

• The slant height (hypotenuse): \( l = 29 \text{ mi} \)
• One leg (half of the base length): \( r = 20 \text{ mi} \) (this represents half of the base length).

Let \( h \) represent the height of the square pyramid. Using the Pythagorean theorem, we have:

\[ l^2 = h^2 + r^2 \]

Substituting the known values:

\[ (29)^2 = h^2 + (20)^2 \]

Calculating \( (29)^2 \) and \( (20)^2 \):

\[ (29)^2 = 841 \] \[ (20)^2 = 400 \]

Now substituting these values back into the equation:

\[ 841 = h^2 + 400 \]

Next, subtract \( 400 \) from both sides:

\[ 841 - 400 = h^2 \] \[ 441 = h^2 \]

Now take the square root of both sides to find the height \( h \):

\[ h = \sqrt{441} = 21 \]

Therefore, the height of the square pyramid is \( \boxed{21} \) miles.