Math / Trigonometry

QuestionA 7 - ft ladder leans against the side of a house. How far Is the bottom of the ladder from the side of the house when the angle of elevation of the ladder Is $64^{\circ}$ ? Round your answer to the nearest tenth. $\square$ ft Check Save For Later Submit Assian

Studdy Solution

STEP 1

What is this asking? How far is the base of a 7-foot ladder from a house if the ladder makes a $64^\circ$ angle with the ground? Watch out! Don't mix up the sides of the triangle!
We're looking for the adjacent side, not the opposite side.

STEP 2

1. Draw a triangle
2. Set up the cosine
3. Solve for the adjacent side

STEP 3

Imagine the ladder, the house, and the ground forming a right triangle!
The ladder is the hypotenuse, the house is the opposite side, and the ground between the ladder and the house is the adjacent side.

STEP 4

Draw a right triangle.
Label the hypotenuse as $7$ ft (the length of the ladder).
Mark the angle between the ladder and the ground as $64^\circ$ .
We want to find the length of the adjacent side, so label it $x$ .

STEP 5

Recall that $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ .
This is perfect because we know the angle ( $\theta = 64^\circ$ ) and the hypotenuse ( $7$ ft), and we want to find the adjacent side!

STEP 6

Substitute the known values into the cosine formula: $\cos(64^\circ) = \frac{x}{7}$

STEP 7

To get $x$ by itself, we need to multiply both sides of the equation by $7$ .
This is like dividing both sides by $\frac{1}{7}$ to turn the $\frac{x}{7}$ into $\frac{x}{7} \cdot \frac{7}{1} = \frac{7x}{7} = \frac{7}{7} \cdot x = 1 \cdot x = x$ .
Remember, we do the same thing to both sides to keep the equation balanced! $7 \cdot \cos(64^\circ) = 7 \cdot \frac{x}{7}$ $7 \cdot \cos(64^\circ) = x$

STEP 8

Now, grab your calculator and find the cosine of $64^\circ$ .
You should get approximately $0.436$ .
Multiply this by $7$ : $x = 7 \cdot \cos(64^\circ) \approx 7 \cdot 0.436 \approx 3.052$

STEP 9

The problem asks for the answer to the nearest tenth, so we round $3.052$ to $3.1$ ft.

STEP 10

The bottom of the ladder is approximately $3.1$ ft from the side of the house.

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QuestionA 7 - ft ladder leans against the side of a house. How far Is the bottom of the ladder from the side of the house when the angle of elevation of the ladder Is $64^{\circ}$ ? Round your answer to the nearest tenth. $\square$ ft Check Save For Later Submit Assian

Studdy Solution

STEP 1

What is this asking? How far is the base of a 7-foot ladder from a house if the ladder makes a $64^\circ$ angle with the ground? Watch out! Don't mix up the sides of the triangle!
We're looking for the adjacent side, not the opposite side.

STEP 2

1. Draw a triangle
2. Set up the cosine
3. Solve for the adjacent side

STEP 3

Imagine the ladder, the house, and the ground forming a right triangle!
The ladder is the hypotenuse, the house is the opposite side, and the ground between the ladder and the house is the adjacent side.

STEP 4

Draw a right triangle.
Label the hypotenuse as $7$ ft (the length of the ladder).
Mark the angle between the ladder and the ground as $64^\circ$ .
We want to find the length of the adjacent side, so label it $x$ .

STEP 5

Recall that $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ .
This is perfect because we know the angle ( $\theta = 64^\circ$ ) and the hypotenuse ( $7$ ft), and we want to find the adjacent side!

STEP 6

Substitute the known values into the cosine formula: $\cos(64^\circ) = \frac{x}{7}$

STEP 7

STEP 8

Now, grab your calculator and find the cosine of $64^\circ$ .
You should get approximately $0.436$ .
Multiply this by $7$ : $x = 7 \cdot \cos(64^\circ) \approx 7 \cdot 0.436 \approx 3.052$

STEP 9

The problem asks for the answer to the nearest tenth, so we round $3.052$ to $3.1$ ft.

STEP 10

The bottom of the ladder is approximately $3.1$ ft from the side of the house.

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QuestionA 7 - ft ladder leans against the side of a house. How far Is the bottom of the ladder from the side of the house when the angle of elevation of the ladder Is 64∘64^{\circ}64∘ ? Round your answer to the nearest tenth. □\square□ ft Check Save For Later Submit Assian

Studdy Solution

STEP 1

What is this asking? How far is the base of a 7-foot ladder from a house if the ladder makes a 64∘64^\circ64∘ angle with the ground? Watch out! Don't mix up the sides of the triangle!We're looking for the *adjacent* side, not the *opposite* side.

STEP 2

1. Draw a triangle2. Set up the cosine3. Solve for the adjacent side

STEP 3

Imagine the ladder, the house, and the ground forming a **right triangle**!The ladder is the **hypotenuse**, the house is the **opposite** side, and the ground between the ladder and the house is the **adjacent** side.

STEP 4

Draw a right triangle.Label the **hypotenuse** as 777 ft (the length of the ladder).Mark the angle between the ladder and the ground as 64∘64^\circ64∘.We want to find the length of the **adjacent** side, so label it xxx.

STEP 5

Recall that cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}cos(θ)=hypotenuseadjacent​.This is perfect because we know the angle (θ=64∘\theta = 64^\circθ=64∘) and the hypotenuse (777 ft), and we want to find the adjacent side!

STEP 6

Substitute the known values into the cosine formula: cos⁡(64∘)=x7 \cos(64^\circ) = \frac{x}{7} cos(64∘)=7x​

STEP 7

STEP 8

Now, grab your calculator and find the cosine of 64∘64^\circ64∘.You should get approximately 0.4360.4360.436.Multiply this by 777: x=7⋅cos⁡(64∘)≈7⋅0.436≈3.052 x = 7 \cdot \cos(64^\circ) \approx 7 \cdot 0.436 \approx 3.052 x=7⋅cos(64∘)≈7⋅0.436≈3.052

STEP 9

The problem asks for the answer to the nearest tenth, so we round 3.0523.0523.052 to **3.13.13.1 ft**.

STEP 10

The bottom of the ladder is approximately 3.13.13.1 ft from the side of the house.

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Start learning now

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QuestionA 7 - ft ladder leans against the side of a house. How far Is the bottom of the ladder from the side of the house when the angle of elevation of the ladder Is $64^{\circ}$ ? Round your answer to the nearest tenth. $\square$ ft Check Save For Later Submit Assian

What is this asking? How far is the base of a 7-foot ladder from a house if the ladder makes a $64^\circ$ angle with the ground? Watch out! Don't mix up the sides of the triangle!
We're looking for the adjacent side, not the opposite side.

1. Draw a triangle
2. Set up the cosine
3. Solve for the adjacent side

Imagine the ladder, the house, and the ground forming a right triangle!
The ladder is the hypotenuse, the house is the opposite side, and the ground between the ladder and the house is the adjacent side.

Draw a right triangle.
Label the hypotenuse as $7$ ft (the length of the ladder).
Mark the angle between the ladder and the ground as $64^\circ$ .
We want to find the length of the adjacent side, so label it $x$ .

Recall that $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ .
This is perfect because we know the angle ( $\theta = 64^\circ$ ) and the hypotenuse ( $7$ ft), and we want to find the adjacent side!

Substitute the known values into the cosine formula: $\cos(64^\circ) = \frac{x}{7}$

Now, grab your calculator and find the cosine of $64^\circ$ .
You should get approximately $0.436$ .
Multiply this by $7$ : $x = 7 \cdot \cos(64^\circ) \approx 7 \cdot 0.436 \approx 3.052$

The problem asks for the answer to the nearest tenth, so we round $3.052$ to $3.1$ ft.

The bottom of the ladder is approximately $3.1$ ft from the side of the house.