To find the longest straw that can fit inside the juice can diagonally, we need to find the diagonal of the cylinder, which is the hypotenuse of a right triangle formed by the height of the cylinder and the diameter of its base. To do that, we can use the Pythagorean Theorem extended to three dimensions.
The formula for the diagonal (d) in three dimensions is similar to the Pythagorean Theorem but includes a third term:
d² = h² + r² + r²
Where:
- d is the diagonal (the length of the longest straw)
- h is the height of the cylinder
- r is the radius of the base of the cylinder
The diameter of the juice can is given as 52 mm, so the radius (r) is half of that:
r = diameter / 2 = 52 mm / 2 = 26 mm
Now plug the height (h = 122 mm) and the radius (r = 26 mm) into the formula:
d² = h² + r² + r²
d² = 122² + 26² + 26²
d² = 14884 + 676 + 676
d² = 16236
To find the diagonal, we take the square root of d²:
d = √16236
This gives us the length of the diagonal in millimeters. But to proceed, let's calculate the square root:
d = √16236 ≈ 127.42 mm
Round the result to the nearest tenth:
d ≈ 127.4 mm
So, the longest straw that can fit inside the juice can diagonally is approximately 127.4 mm.
The Pythagorean Theorem in Three Dimensions
Math 8th Pre-AlgebraTriangles
A juice can is in the shape of a cylinder. It has a height of 122 mm and a diameter of 52 mm. What is the longest straw that can fit completely inside the juice can diagonally? Round the answer to the nearest tenth. (1 point)
5 answers
wrong
Apologies for the confusion. Let's correct that.
To find the longest straw that can fit inside the juice can diagonally, we're looking for the length of the diagonal within the cylinder. This length will pass from one edge of the circular base, through the center of the cylinder, to the top edge of the opposite side of the base. Since the diameter provides the longest straight-line distance across the circle (i.e., the cylinder base), we can consider the diameter and the height of the cylinder as the legs of a right-angled triangle.
The Pythagorean Theorem in three dimensions in terms of a cylinder is:
\[ d^2 = h^2 + (diameter)^2 \]
Given:
Height (h) = 122 mm
Diameter of the base = 52 mm, which means the radius (r) is not needed in this case.
Let's calculate the length of the diagonal (d), which will be the longest straw:
\[ d^2 = 122^2 + 52^2 \]
\[ d^2 = 14884 + 2704 \]
\[ d^2 = 17588 \]
Now take the square root of both sides to solve for d:
\[ d = \sqrt{17588} \]
\[ d \approx 132.6 \] mm
Rounded to the nearest tenth:
\[ d \approx 132.7 \] mm
Therefore, the longest straw that can fit inside the juice can diagonally is approximately 132.7 mm.
To find the longest straw that can fit inside the juice can diagonally, we're looking for the length of the diagonal within the cylinder. This length will pass from one edge of the circular base, through the center of the cylinder, to the top edge of the opposite side of the base. Since the diameter provides the longest straight-line distance across the circle (i.e., the cylinder base), we can consider the diameter and the height of the cylinder as the legs of a right-angled triangle.
The Pythagorean Theorem in three dimensions in terms of a cylinder is:
\[ d^2 = h^2 + (diameter)^2 \]
Given:
Height (h) = 122 mm
Diameter of the base = 52 mm, which means the radius (r) is not needed in this case.
Let's calculate the length of the diagonal (d), which will be the longest straw:
\[ d^2 = 122^2 + 52^2 \]
\[ d^2 = 14884 + 2704 \]
\[ d^2 = 17588 \]
Now take the square root of both sides to solve for d:
\[ d = \sqrt{17588} \]
\[ d \approx 132.6 \] mm
Rounded to the nearest tenth:
\[ d \approx 132.7 \] mm
Therefore, the longest straw that can fit inside the juice can diagonally is approximately 132.7 mm.
incorrect
I apologize for the oversight. Let us try again.
To find the longest straw that can fit inside the juice can diagonally, we must calculate the length of the diagonal that spans from the bottom of the can, passes through the center, and reaches to the top edge of the can, exactly opposite from where it began. This diagonal forms the hypotenuse of a right-angled triangle where the height of the can is one leg and the diameter of the base is the other leg.
The formula for the length of this diagonal (D) can be derived from the Pythagorean theorem and in the case of a cylinder, is:
\[ D^2 = h^2 + d^2 \]
where:
- \( D \) is the diagonal (the length of the longest straw),
- \( h \) is the height of the cylinder, and
- \( d \) is the diameter of the base of the cylinder.
Now, let's plug in the height (\( h = 122 \) mm) and the diameter (\( d = 52 \) mm) into the formula.
\[ D^2 = 122^2 + 52^2 \]
\[ D^2 = 14884 + 2704 \]
\[ D^2 = 17588 \]
Find the diagonal by taking the square root of \( D^2 \):
\[ D = \sqrt{17588} \]
\[ D \approx 132.6 \] mm
Since we need the answer to the nearest tenth, the diagonal \( D \) is approximately:
\[ D \approx 132.6 \] mm
Therefore, the longest straw that can fit inside the juice can diagonally to the nearest tenth is 132.6 mm.
If this response still does not satisfy the question, please double-check the details of the problem as I've used the standard approach for three-dimensional Pythagorean calculations.
To find the longest straw that can fit inside the juice can diagonally, we must calculate the length of the diagonal that spans from the bottom of the can, passes through the center, and reaches to the top edge of the can, exactly opposite from where it began. This diagonal forms the hypotenuse of a right-angled triangle where the height of the can is one leg and the diameter of the base is the other leg.
The formula for the length of this diagonal (D) can be derived from the Pythagorean theorem and in the case of a cylinder, is:
\[ D^2 = h^2 + d^2 \]
where:
- \( D \) is the diagonal (the length of the longest straw),
- \( h \) is the height of the cylinder, and
- \( d \) is the diameter of the base of the cylinder.
Now, let's plug in the height (\( h = 122 \) mm) and the diameter (\( d = 52 \) mm) into the formula.
\[ D^2 = 122^2 + 52^2 \]
\[ D^2 = 14884 + 2704 \]
\[ D^2 = 17588 \]
Find the diagonal by taking the square root of \( D^2 \):
\[ D = \sqrt{17588} \]
\[ D \approx 132.6 \] mm
Since we need the answer to the nearest tenth, the diagonal \( D \) is approximately:
\[ D \approx 132.6 \] mm
Therefore, the longest straw that can fit inside the juice can diagonally to the nearest tenth is 132.6 mm.
If this response still does not satisfy the question, please double-check the details of the problem as I've used the standard approach for three-dimensional Pythagorean calculations.