Math / Trigonometry

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Bo is flying a kite, holding his hands a distance of 3.5 feet above the ground and letting all the kite's string play out. He measures the angle of elevation from his hand to the kite to be $29^{\circ}$ . If the string from the kite to his hand is 110 feet long, how many feet is the kite above the ground? Round your answer to the nearest tenth of a foot if necessary.
Answer Attempt 1 out of 2 $\qquad$ feet Submit Answer

Studdy Solution

STEP 1

What is this asking? How high is a kite if a person is holding the string 3.5 feet off the ground, the string is 110 feet long, and the string makes a 29-degree angle with the ground? Watch out! Don't forget to add the initial height of the hand holding the string!

STEP 2

1. Visualize the problem
2. Find the kite's height relative to the hand
3. Find the kite's total height

STEP 3

Alright, imagine a right triangle!
The string forms the hypotenuse, the height of the kite above the hand is the opposite side, and the distance along the ground is the adjacent side.
The angle between the string and the ground is our 29-degree angle.

STEP 4

We can use the sine function because we know the hypotenuse ( $110$ feet) and the angle ( $29^\circ$ ), and we want to find the opposite side (the height above the hand).
Remember, sine is defined as the ratio of the opposite side to the hypotenuse.

STEP 5

So, we have $\sin(29^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}$ .
Let's call the opposite side $h_1$ , which is the height of the kite above the hand.
Plugging in our values, we get $\sin(29^\circ) = \frac{h_1}{110}$ .

STEP 6

To solve for $h_1$ , we multiply both sides of the equation by $110$ : $110 \cdot \sin(29^\circ) = 110 \cdot \frac{h_1}{110}$ .
This simplifies to $h_1 = 110 \cdot \sin(29^\circ)$ .

STEP 7

Now, let's calculate $h_1$ : $h_1 = 110 \cdot \sin(29^\circ) \approx 110 \cdot 0.4848 \approx 53.33$ feet.
So, the kite is approximately 53.33 feet above the hand holding the string.

STEP 8

We know the kite is $53.33$ feet above the hand, and the hand is $3.5$ feet above the ground.
To find the total height, we simply add these two heights together.

STEP 9

Let's call the total height $h$ .
So, $h = h_1 + 3.5$ .
Plugging in the value we found for $h_1$ , we get $h = 53.33 + 3.5 = 56.83$ feet.

STEP 10

Rounding to the nearest tenth gives us a final height of $56.8$ feet.

STEP 11

The kite is approximately $56.8$ feet above the ground.

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Studdy Solution

STEP 1

What is this asking? How high is a kite if a person is holding the string 3.5 feet off the ground, the string is 110 feet long, and the string makes a 29-degree angle with the ground? Watch out! Don't forget to add the initial height of the hand holding the string!

STEP 2

1. Visualize the problem
2. Find the kite's height relative to the hand
3. Find the kite's total height

STEP 3

Alright, imagine a right triangle!
The string forms the hypotenuse, the height of the kite above the hand is the opposite side, and the distance along the ground is the adjacent side.
The angle between the string and the ground is our 29-degree angle.

STEP 4

We can use the sine function because we know the hypotenuse ( $110$ feet) and the angle ( $29^\circ$ ), and we want to find the opposite side (the height above the hand).
Remember, sine is defined as the ratio of the opposite side to the hypotenuse.

STEP 5

So, we have $\sin(29^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}$ .
Let's call the opposite side $h_1$ , which is the height of the kite above the hand.
Plugging in our values, we get $\sin(29^\circ) = \frac{h_1}{110}$ .

STEP 6

To solve for $h_1$ , we multiply both sides of the equation by $110$ : $110 \cdot \sin(29^\circ) = 110 \cdot \frac{h_1}{110}$ .
This simplifies to $h_1 = 110 \cdot \sin(29^\circ)$ .

STEP 7

Now, let's calculate $h_1$ : $h_1 = 110 \cdot \sin(29^\circ) \approx 110 \cdot 0.4848 \approx 53.33$ feet.
So, the kite is approximately 53.33 feet above the hand holding the string.

STEP 8

We know the kite is $53.33$ feet above the hand, and the hand is $3.5$ feet above the ground.
To find the total height, we simply add these two heights together.

STEP 9

Let's call the total height $h$ .
So, $h = h_1 + 3.5$ .
Plugging in the value we found for $h_1$ , we get $h = 53.33 + 3.5 = 56.83$ feet.

STEP 10

Rounding to the nearest tenth gives us a final height of $56.8$ feet.

STEP 11

The kite is approximately $56.8$ feet above the ground.

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Studdy Solution

STEP 1

What is this asking? How high is a kite if a person is holding the string 3.5 feet off the ground, the string is 110 feet long, and the string makes a 29-degree angle with the ground? Watch out! Don't forget to add the initial height of the hand holding the string!

STEP 2

1. Visualize the problem2. Find the kite's height relative to the hand3. Find the kite's total height

STEP 3

Alright, imagine a right triangle!The string forms the **hypotenuse**, the height of the kite *above the hand* is the **opposite side**, and the distance along the ground is the **adjacent side**.The angle between the string and the ground is our **29-degree angle**.

STEP 4

We can use the sine function because we know the hypotenuse (110110110 feet) and the angle (29∘29^\circ29∘), and we want to find the opposite side (the height above the hand).Remember, sine is defined as the ratio of the **opposite side** to the **hypotenuse**.

STEP 5

STEP 6

To solve for h1h_1h1​, we **multiply both sides** of the equation by 110110110: 110⋅sin⁡(29∘)=110⋅h1110110 \cdot \sin(29^\circ) = 110 \cdot \frac{h_1}{110}110⋅sin(29∘)=110⋅110h1​​.This simplifies to h1=110⋅sin⁡(29∘)h_1 = 110 \cdot \sin(29^\circ)h1​=110⋅sin(29∘).

STEP 7

Now, let's **calculate** h1h_1h1​: h1=110⋅sin⁡(29∘)≈110⋅0.4848≈53.33h_1 = 110 \cdot \sin(29^\circ) \approx 110 \cdot 0.4848 \approx 53.33h1​=110⋅sin(29∘)≈110⋅0.4848≈53.33 feet.So, the kite is approximately **53.33 feet** above the hand holding the string.

STEP 8

We know the kite is 53.3353.3353.33 feet above the hand, and the hand is 3.53.53.5 feet above the ground.To find the total height, we simply **add** these two heights together.

STEP 9

Let's call the total height hhh.So, h=h1+3.5h = h_1 + 3.5h=h1​+3.5.Plugging in the value we found for h1h_1h1​, we get h=53.33+3.5=56.83h = 53.33 + 3.5 = 56.83h=53.33+3.5=56.83 feet.

STEP 10

Rounding to the nearest tenth gives us a **final height** of 56.856.856.8 feet.

STEP 11

The kite is approximately 56.856.856.8 feet above the ground.

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1. Visualize the problem
2. Find the kite's height relative to the hand
3. Find the kite's total height

Alright, imagine a right triangle!
The string forms the hypotenuse, the height of the kite above the hand is the opposite side, and the distance along the ground is the adjacent side.
The angle between the string and the ground is our 29-degree angle.

We can use the sine function because we know the hypotenuse ( $110$ feet) and the angle ( $29^\circ$ ), and we want to find the opposite side (the height above the hand).
Remember, sine is defined as the ratio of the opposite side to the hypotenuse.

To solve for $h_1$ , we multiply both sides of the equation by $110$ : $110 \cdot \sin(29^\circ) = 110 \cdot \frac{h_1}{110}$ .
This simplifies to $h_1 = 110 \cdot \sin(29^\circ)$ .

Now, let's calculate $h_1$ : $h_1 = 110 \cdot \sin(29^\circ) \approx 110 \cdot 0.4848 \approx 53.33$ feet.
So, the kite is approximately 53.33 feet above the hand holding the string.

We know the kite is $53.33$ feet above the hand, and the hand is $3.5$ feet above the ground.
To find the total height, we simply add these two heights together.

Let's call the total height $h$ .
So, $h = h_1 + 3.5$ .
Plugging in the value we found for $h_1$ , we get $h = 53.33 + 3.5 = 56.83$ feet.

Rounding to the nearest tenth gives us a final height of $56.8$ feet.

The kite is approximately $56.8$ feet above the ground.