QuestionA cylinder has a diameter of 10 in and height of 13 in. Find $\theta$ using: A. $\cos (\theta)=\frac{10}{13}$ , B. $\sin (\theta)=\frac{10}{13}$ , C. $\sin (\theta)=\frac{13}{10}$ , D. 10.

Studdy Solution

STEP 1

Assumptions1. The diameter of the cylinder is10 inches. . The height of the cylinder is13 inches.
3. $\theta$ is the angle formed between the base of the cylinder and the longest line segment joining any two points in the cylinder. This longest line segment is the slant height of the cylinder, which is the hypotenuse of the right triangle formed by the radius, height, and slant height of the cylinder.

STEP 2

We need to find the relationship between the diameter, height, and the angle $\theta$ . To do this, we first need to understand that the longest line segment joining any two points in the cylinder is the slant height, which forms a right triangle with the radius and the height of the cylinder.

STEP 3

Since the diameter of the cylinder is10 inches, the radius of the cylinder is half of the diameter, which is5 inches.
$Radius = \frac{iameter}{2} = \frac{10}{2} =5\, inches$

STEP 4

In the right triangle formed by the radius, height, and slant height of the cylinder, the radius and height are the two legs, and the slant height is the hypotenuse. The angle $\theta$ is formed between the base (radius) and the hypotenuse (slant height).

STEP 5

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side (in this case, the radius) to the hypotenuse (in this case, the slant height). Therefore, we can use the cosine function to find the relationship between the radius, height, and the angle $\theta$ .
$\cos(\theta) = \frac{Radius}{Slant\, height}$

STEP 6

However, we don't know the slant height. But we can calculate it using the Pythagorean theorem since we know the radius and the height of the cylinder. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (slant height) is equal to the sum of the squares of the lengths of the other two sides (radius and height).
$Slant\, height = \sqrt{Radius^2 + Height^2}$

STEP 7

Plug in the values for the radius and the height to calculate the slant height.
$Slant\, height = \sqrt{5^2 +13^2}$

STEP 8

Calculate the slant height.
$Slant\, height = \sqrt{25 +169} = \sqrt{194}$

STEP 9

Now that we have the slant height, we can substitute it back into the equation for cosine.
$\cos(\theta) = \frac{5}{\sqrt{194}}$

STEP 10

implify the equation.
$\cos(\theta) = \frac{5\sqrt{194}}{194}$ None of the given options match this result, so there seems to be a mistake in the problem. The correct equation to find $\theta$ should be $\cos(\theta) = \frac{5\sqrt{194}}{194}$ .

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QuestionA cylinder has a diameter of 10 in and height of 13 in. Find $\theta$ using: A. $\cos (\theta)=\frac{10}{13}$ , B. $\sin (\theta)=\frac{10}{13}$ , C. $\sin (\theta)=\frac{13}{10}$ , D. 10.

Studdy Solution

STEP 1

STEP 2

We need to find the relationship between the diameter, height, and the angle $\theta$ . To do this, we first need to understand that the longest line segment joining any two points in the cylinder is the slant height, which forms a right triangle with the radius and the height of the cylinder.

STEP 3

Since the diameter of the cylinder is10 inches, the radius of the cylinder is half of the diameter, which is5 inches.
$Radius = \frac{iameter}{2} = \frac{10}{2} =5\, inches$

STEP 4

In the right triangle formed by the radius, height, and slant height of the cylinder, the radius and height are the two legs, and the slant height is the hypotenuse. The angle $\theta$ is formed between the base (radius) and the hypotenuse (slant height).

STEP 5

STEP 6

STEP 7

Plug in the values for the radius and the height to calculate the slant height.
$Slant\, height = \sqrt{5^2 +13^2}$

STEP 8

Calculate the slant height.
$Slant\, height = \sqrt{25 +169} = \sqrt{194}$

STEP 9

Now that we have the slant height, we can substitute it back into the equation for cosine.
$\cos(\theta) = \frac{5}{\sqrt{194}}$

STEP 10

implify the equation.
$\cos(\theta) = \frac{5\sqrt{194}}{194}$ None of the given options match this result, so there seems to be a mistake in the problem. The correct equation to find $\theta$ should be $\cos(\theta) = \frac{5\sqrt{194}}{194}$ .

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QuestionA cylinder has a diameter of 10 in and height of 13 in. Find θ\thetaθ using: A. cos⁡(θ)=1013\cos (\theta)=\frac{10}{13}cos(θ)=1310​, B. sin⁡(θ)=1013\sin (\theta)=\frac{10}{13}sin(θ)=1310​, C. sin⁡(θ)=1310\sin (\theta)=\frac{13}{10}sin(θ)=1013​, D. 10.

Studdy Solution

STEP 1

STEP 2

We need to find the relationship between the diameter, height, and the angle θ\thetaθ. To do this, we first need to understand that the longest line segment joining any two points in the cylinder is the slant height, which forms a right triangle with the radius and the height of the cylinder.

STEP 3

Since the diameter of the cylinder is10 inches, the radius of the cylinder is half of the diameter, which is5 inches.Radius=iameter2=102=5 inchesRadius = \frac{iameter}{2} = \frac{10}{2} =5\, inchesRadius=2iameter​=210​=5inches

STEP 4

In the right triangle formed by the radius, height, and slant height of the cylinder, the radius and height are the two legs, and the slant height is the hypotenuse. The angle θ\thetaθ is formed between the base (radius) and the hypotenuse (slant height).

STEP 5

STEP 6

STEP 7

Plug in the values for the radius and the height to calculate the slant height.Slant height=52+132Slant\, height = \sqrt{5^2 +13^2}Slantheight=52+132​

STEP 8

Calculate the slant height.Slant height=25+169=194Slant\, height = \sqrt{25 +169} = \sqrt{194}Slantheight=25+169​=194​

STEP 9

Now that we have the slant height, we can substitute it back into the equation for cosine.cos⁡(θ)=5194\cos(\theta) = \frac{5}{\sqrt{194}}cos(θ)=194​5​

STEP 10

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QuestionA cylinder has a diameter of 10 in and height of 13 in. Find $\theta$ using: A. $\cos (\theta)=\frac{10}{13}$ , B. $\sin (\theta)=\frac{10}{13}$ , C. $\sin (\theta)=\frac{13}{10}$ , D. 10.

We need to find the relationship between the diameter, height, and the angle $\theta$ . To do this, we first need to understand that the longest line segment joining any two points in the cylinder is the slant height, which forms a right triangle with the radius and the height of the cylinder.

Since the diameter of the cylinder is10 inches, the radius of the cylinder is half of the diameter, which is5 inches.
$Radius = \frac{iameter}{2} = \frac{10}{2} =5\, inches$

In the right triangle formed by the radius, height, and slant height of the cylinder, the radius and height are the two legs, and the slant height is the hypotenuse. The angle $\theta$ is formed between the base (radius) and the hypotenuse (slant height).

Plug in the values for the radius and the height to calculate the slant height.
$Slant\, height = \sqrt{5^2 +13^2}$

Calculate the slant height.
$Slant\, height = \sqrt{25 +169} = \sqrt{194}$

Now that we have the slant height, we can substitute it back into the equation for cosine.
$\cos(\theta) = \frac{5}{\sqrt{194}}$