Question

Chem Tech makes paper cones used to filter liquids in lab experiments. To make the paper cones, the company starts with the net of the shape, where the slant height is needed. The company wants the height of the cones to be 6 cm. If they make the diameter 13 cm, what must the slant height be? Use the Pythagorean Theorem and round the answer to the nearest tenth

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the slant height, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the height of the cone is one of the legs of the right triangle, and the radius (half the diameter) of the base is the other leg. The slant height is the hypotenuse.

Using the Pythagorean theorem:

(slant height)^2 = (height)^2 + (radius)^2

Since the height is given as 6 cm and the radius is half the diameter (13 cm), we can substitute these values into the equation:

(slant height)^2 = (6 cm)^2 + (6.5 cm)^2

(slant height)^2 = 36 cm^2 + 42.25 cm^2

(slant height)^2 = 78.25 cm^2

Taking the square root of both sides to solve for the slant height:

slant height = √78.25 cm^2

slant height ≈ 8.8 cm

Therefore, the slant height of the paper cones should be rounded to the nearest tenth: 8.8 cm.

Answer 2

A circus tent is in the shape of a square pyramid. Each edge of the square base is 20 m and its slant height is 13.5 m. Find the height of the tent. Round the answer to the nearest tenth.

Answer 3

[AI-generated answer — Bot GPT 3.5]

To find the height of the tent, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the height of the tent is one of the legs of the right triangle, and half the diagonal of the square base is the other leg. The slant height is the hypotenuse.

Using the Pythagorean theorem:

(slant height)^2 = (height)^2 + (half diagonal)^2

Since the slant height is given as 13.5 m and the side length of the square base is 20 m, we can calculate half the diagonal of the square base using the Pythagorean theorem for right triangles with sides of length 20 m:

(diagonal)^2 = (side length)^2 + (side length)^2

(diagonal)^2 = (20 m)^2 + (20 m)^2

(diagonal)^2 = 400 m^2 + 400 m^2

(diagonal)^2 = 800 m^2

(diagonal) = √800 m

(diagonal) ≈ 28.3 m

Therefore, half the diagonal of the square base is (28.3 m) / 2 = 14.15 m.

Now we can substitute the values into the equation to find the height of the tent:

(13.5 m)^2 = (height)^2 + (14.15 m)^2

182.25 m^2 = (height)^2 + 199.9225 m^2

(height)^2 = 182.25 m^2 - 199.9225 m^2

(height)^2 = -17.6725 m^2

Since we are finding the height, we can ignore the negative root and take the positive root:

height = √(-17.6725 m^2)

Since the square root of a negative number is not a real number, there is an error in the initial problem statement. The given measurements of the square base and the slant height do not form a valid right triangle. Therefore, we cannot determine the height of the tent.

Answer 4

[AI-generated answer — Explain Bot]

To find the slant height of the paper cone, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the height of the cone (6 cm) is one of the legs of the right triangle. The radius of the cone is half the diameter, so it is 13 cm divided by 2, which is 6.5 cm. This radius will be the other leg of the right triangle.

Let's label the slant height of the cone as 's'. According to the Pythagorean Theorem, we have:

s² = (radius)² + (height)²
s² = 6.5² + 6²
s² = 42.25 + 36
s² ≈ 78.25

To find the slant height, we take the square root of both sides of the equation:

s ≈ √78.25
s ≈ 8.8 cm (rounded to the nearest tenth)

Therefore, the slant height of the paper cone should be approximately 8.8 cm.