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.
To find the slant height of the cone, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) 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 sides, the radius (half the diameter) of the base is the other side, and the slant height is the hypotenuse.
Using the Pythagorean Theorem equation, we have:
slant height^2 = height^2 + radius^2
plugging in the given values, we get:
slant height^2 = 6^2 + (13/2)^2
slant height^2 = 36 + 169/4
slant height^2 = 144/4 + 169/4
slant height^2 = 313/4
Now, we can solve this equation for the slant height:
slant height = sqrt(313/4)
slant height ≈ sqrt(78.25)
slant height ≈ 8.8 cm
Therefore, the slant height of the cone should be approximately 8.8 cm.
To find the slant height of the 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 diameter of the cone is 13 cm, which means the radius is half of the diameter, or 13/2 = 6.5 cm.
Let's call the slant height "s", the height "h", and the radius "r".
We are given that the height of the cone is 6 cm (h = 6 cm) and the radius is 6.5 cm (r = 6.5 cm).
Using the Pythagorean Theorem, we can write the equation as:
s^2 = r^2 + h^2
Substituting the values we have:
s^2 = (6.5)^2 + (6)^2
s^2 = 42.25 + 36
s^2 = 78.25
To find the slant height "s", we take the square root of both sides:
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.
To solve this problem, we will 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 slant height of the cone will be the hypotenuse, and the height and the radius of the base will be the other two sides. Here's how we can solve the problem step-by-step:
1. Start by drawing a diagram to visualize the problem. Draw a right triangle with the height (h) as one leg, the radius (r) as the other leg, and the slant height (s) as the hypotenuse. Label the height as 6 cm and the radius as half of the diameter, which is 13 cm divided by 2, equal to 6.5 cm.
2. Apply the Pythagorean Theorem equation: s^2 = h^2 + r^2
3. Substitute the given values into the equation:
s^2 = 6^2 + 6.5^2
s^2 = 36 + 42.25
s^2 ≈ 78.25
4. Take the square root of both sides of the equation to solve for s:
s ≈ √78.25
s ≈ 8.8
Therefore, the slant height of the paper cone should be approximately 8.8 cm when the height is 6 cm and the diameter is 13 cm.