A drawing shows a circle showing diameter on top of a sector of a circle showing slant height. A double-headed arrow crossing the center of the circle is labeled as d equals diameter. A dotted double-headed arrow is drawn from the top point of the sector of the circle to a point along the curved edge. The arrow is labeled as lower L equals slant height.
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.
The Pythagorean theorem states that in a right triangle, the square of the length 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 base of the cone is the hypotenuse, and the height of the cone is one of the other sides. We want to find the length of the slant height, which is the other side.
Using the Pythagorean theorem, we have:
diameter^2 = height^2 + slant height^2
Substituting the given values, we have:
13^2 = 6^2 + slant height^2
169 = 36 + slant height^2
133 = slant height^2
Taking the square root of both sides, we have:
slant height = √133 ≈ 11.5 cm
Therefore, the slant height of the cone should be approximately 11.5 cm.
To calculate the slant height of the cone, we can use the Pythagorean Theorem. The Pythagorean Theorem states that for 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 base of the cone is 13 cm, which means the radius is half of the diameter, or 6.5 cm.
We can create a right triangle using the diameter as the base, the radius as one of the legs, and the slant height as the hypotenuse.
Using the Pythagorean theorem, we get:
Hypotenuse^2 = Base^2 + Height^2
Slant Height^2 = Radius^2 + Height^2
Slant Height^2 = 6.5^2 + 6^2
Slant Height^2 = 42.25 + 36
Slant Height^2 = 78.25
Taking the square root of both sides, we find:
Slant Height ≈ √78.25
Slant Height ≈ 8.8 cm (rounded to the nearest tenth)
Therefore, the slant height of the cone should be approximately 8.8 cm.
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 diameter of the circle represents one of the sides of the right triangle, and the slant height represents the hypotenuse.
We know that the height of the cone is 6 cm, and the diameter is 13 cm. To find the slant height, we need to find the length of the third side of the right triangle, which we can call "b."
Using the Pythagorean Theorem, we have the following equation:
b^2 = (d/2)^2 + h^2
where d is the diameter (13 cm) and h is the height (6 cm). Plugging in the values, we get:
b^2 = (13/2)^2 + 6^2
b^2 = (6.5)^2 + 36
b^2 = 42.25 + 36
b^2 = 78.25
To find the value of b (the slant height), we take the square root of both sides of the equation:
b = sqrt(78.25)
Evaluating this expression, we find that the slant height is approximately 8.85 cm (rounded to the nearest tenth).