Find the length of the diagonal of the prism.

Question

(1 point)

PRISM: 12 cm tall, 3cm wide, 4 cm long

Answer 1

To find the length of the diagonal of a rectangular prism, we can use the formula for the space diagonal \(d\):

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where \(l\) is the length, \(w\) is the width, and \(h\) is the height of the prism.

Given:

• Height \(h = 12 , \text{cm}\)
• Width \(w = 3 , \text{cm}\)
• Length \(l = 4 , \text{cm}\)

Now we substitute the values into the formula:

\[ d = \sqrt{(4)^2 + (3)^2 + (12)^2} \] \[ d = \sqrt{16 + 9 + 144} \] \[ d = \sqrt{169} \] \[ d = 13 , \text{cm} \]

Therefore, the length of the diagonal of the prism is \(13 , \text{cm}\).

Answer 2

The Great Pyramid of Giza in Egypt has a square base with edge length of 230.5 m and a slant height
of 180.2 m. Find the height of the pyramid. Round the answer to the nearest tenth.(1 point)

Answer 3

To find the height of the Great Pyramid of Giza, we can use the relationship between the slant height, the height, and half the base length.

The pyramid has a square base, so:

• The edge length of the base is \(230.5 , \text{m}\).
• The slant height \(l\) is \(180.2 , \text{m}\).

The half-length of the base \(b\) is:

\[ b = \frac{230.5}{2} = 115.25 , \text{m} \]

We can use the Pythagorean theorem to find the height \(h\) of the pyramid:

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

Rearranging the formula to solve for \(h\) gives:

\[ h^2 = l^2 - b^2 \]

Now substitute in the known values:

\[ h^2 = (180.2)^2 - (115.25)^2 \]

Calculating \( (180.2)^2 \) and \( (115.25)^2 \):

\[ (180.2)^2 = 32472.04 \] \[ (115.25)^2 = 13273.5625 \]

Now, substitute these values into the equation:

\[ h^2 = 32472.04 - 13273.5625 \]

Calculating the difference:

\[ h^2 = 19198.4775 \]

Taking the square root to find \(h\):

\[ h = \sqrt{19198.4775} \approx 138.7 , \text{m} \]

Rounding to the nearest tenth, the height of the pyramid is approximately 138.7 m.

Answer 4

Keysha is building a coffee table. It’s going to be in the shape of a rectangular prism, without the
faces. She wants to make the table decorative by adding metal diagonals. The length of the coffee
table will be 44 in., the width 24 in., and the height 14 in. Find the length Keysha should make each
diagonal. Round the answer to the nearest tenth.(1 point)

Answer 5

To find the length of the diagonal of a rectangular prism (in this case, the coffee table that Keysha is building), we can use the formula for the space diagonal \(d\):

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where:

• \(l\) is the length,
• \(w\) is the width,
• \(h\) is the height.

Given:

• Length \(l = 44 , \text{in}\)
• Width \(w = 24 , \text{in}\)
• Height \(h = 14 , \text{in}\)

Now we substitute these values into the formula:

\[ d = \sqrt{(44)^2 + (24)^2 + (14)^2} \]

Calculating each term:

\[ (44)^2 = 1936 \] \[ (24)^2 = 576 \] \[ (14)^2 = 196 \]

Now, summing these values:

\[ d = \sqrt{1936 + 576 + 196} \] \[ d = \sqrt{2708} \]

Calculating the square root:

\[ d \approx 52.0 , \text{in} \]

Thus, rounding to the nearest tenth, Keysha should make each diagonal approximately 52.0 inches long.