Use the image to answer the question.
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.
(1 point)
Responses
2.5 cm
2.5 cm
8.8 cm
8.8 cm
11.5 cm
11.5 cm
14.3 cm
14.3 cm
To find the slant height, we can use the Pythagorean theorem. The height of the cone is given as 6 cm and the diameter is given as 13 cm.
Let's label the slant height as 'x'. Now, we can use the Pythagorean theorem:
x^2 = (6 cm)^2 + (13 cm/2)^2
x^2 = 36 cm^2 + 42.25 cm^2
x^2 = 78.25 cm^2
Taking the square root of both sides:
x ≈ 8.8 cm
Therefore, the slant height should be approximately 8.8 cm.
To answer this question, 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 hypotenuse is the slant height, the longer side of the right triangle. The other two sides are the height and the radius (half of the diameter), forming a right triangle within the sector of the circle.
Given that the height of the cone is 6 cm and the diameter is 13 cm, we can find the radius by dividing the diameter by 2: radius = 13 cm / 2 = 6.5 cm.
Using the Pythagorean Theorem, we can set up the equation:
slant_height^2 = height^2 + radius^2
Substituting the known values:
slant_height^2 = 6^2 + 6.5^2
Simplifying:
slant_height^2 = 36 + 42.25
slant_height^2 = 78.25
To find the slant height, we take the square root of both sides:
slant_height = √(78.25)
Using a calculator, we find that the square root of 78.25 is approximately 8.8 cm.
Therefore, the slant height of the paper 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 base of the cone (d) is the base of the right triangle, and the slant height (L) is the hypotenuse.
Given:
Height of the cone (h) = 6 cm
Diameter of the base (d) = 13 cm
The radius of the base (r) is half of the diameter, so r = d/2 = 13/2 = 6.5 cm.
We can now use the Pythagorean Theorem:
L^2 = r^2 + h^2
L^2 = (6.5)^2 + (6)^2
L^2 = 42.25 + 36
L^2 = 78.25
Taking the square root of both sides:
L = √78.25
L ≈ 8.8 cm
Therefore, the slant height of the cone should be approximately 8.8 cm.