QuestionA ladder leans against a wall at $60^{\circ}$ . The base is $2 \mathrm{~m}$ from the wall. Find the wall height and ladder length. $[\sin 60=0.866, \cos 60=0.5, \tan 60=1.732]$

Studdy Solution

STEP 1

Assumptions1. The ladder makes an angle of $60^{\circ}$ with the horizontal. . The distance from the foot of the ladder to the wall is $\mathrm{~m}$ .
3. We are given the trigonometric values for $\sin60^{\circ}$ , $\cos60^{\circ}$ , and $\tan60^{\circ}$ .
4. We are assuming the wall and the ground meet at a right angle (90 degrees).

STEP 2

We'll first find the height of the wall. We can use the trigonometric relationship of sine, which is defined as the ratio of the opposite side (height of the wall in this case) to the hypotenuse (length of the ladder). $\sin(\theta) = \frac{opposite}{hypotenuse}$

STEP 3

But we don't know the length of the ladder (hypotenuse) yet, so we'll use the tangent of the angle instead, which is the ratio of the opposite side (height of the wall) to the adjacent side (distance from the foot of the ladder to the wall).
$\tan(\theta) = \frac{opposite}{adjacent}$

STEP 4

Now, plug in the given values for $\theta$ and the adjacent side to calculate the height of the wall.
$\tan(60^{\circ}) = \frac{height}{2 \mathrm{~m}}$

STEP 5

Rearrange the equation to solve for the height.
$height = \tan(60^{\circ}) \times2 \mathrm{~m}$

STEP 6

Substitute the given value for $\tan(60^{\circ})$ into the equation.
$height =1.732 \times2 \mathrm{~m}$

STEP 7

Calculate the height of the wall.
$height =1.732 \times2 \mathrm{~m} =3.464 \mathrm{~m}$

STEP 8

Next, we'll find the length of the ladder. We can use the trigonometric relationship of cosine, which is defined as the ratio of the adjacent side (distance from the foot of the ladder to the wall) to the hypotenuse (length of the ladder).
$\cos(\theta) = \frac{adjacent}{hypotenuse}$

STEP 9

Now, plug in the given values for $\theta$ and the adjacent side to calculate the length of the ladder.
$\cos(60^{\circ}) = \frac{2 \mathrm{~m}}{length}$

STEP 10

Rearrange the equation to solve for the length.
$length = \frac{2 \mathrm{~m}}{\cos(60^{\circ})}$

STEP 11

Substitute the given value for $\cos(60^{\circ})$ into the equation.
$length = \frac{ \mathrm{~m}}{0.5}$

STEP 12

Calculate the length of the ladder.
$length = \frac{2 \mathrm{~m}}{0.5} =4 \mathrm{~m}$ So, the height of the wall is $.464 \mathrm{~m}$ and the length of the ladder is $4 \mathrm{~m}$ .

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QuestionA ladder leans against a wall at $60^{\circ}$ . The base is $2 \mathrm{~m}$ from the wall. Find the wall height and ladder length. $[\sin 60=0.866, \cos 60=0.5, \tan 60=1.732]$

Studdy Solution

STEP 1

STEP 2

We'll first find the height of the wall. We can use the trigonometric relationship of sine, which is defined as the ratio of the opposite side (height of the wall in this case) to the hypotenuse (length of the ladder). $\sin(\theta) = \frac{opposite}{hypotenuse}$

STEP 3

But we don't know the length of the ladder (hypotenuse) yet, so we'll use the tangent of the angle instead, which is the ratio of the opposite side (height of the wall) to the adjacent side (distance from the foot of the ladder to the wall).
$\tan(\theta) = \frac{opposite}{adjacent}$

STEP 4

Now, plug in the given values for $\theta$ and the adjacent side to calculate the height of the wall.
$\tan(60^{\circ}) = \frac{height}{2 \mathrm{~m}}$

STEP 5

Rearrange the equation to solve for the height.
$height = \tan(60^{\circ}) \times2 \mathrm{~m}$

STEP 6

Substitute the given value for $\tan(60^{\circ})$ into the equation.
$height =1.732 \times2 \mathrm{~m}$

STEP 7

Calculate the height of the wall.
$height =1.732 \times2 \mathrm{~m} =3.464 \mathrm{~m}$

STEP 8

Next, we'll find the length of the ladder. We can use the trigonometric relationship of cosine, which is defined as the ratio of the adjacent side (distance from the foot of the ladder to the wall) to the hypotenuse (length of the ladder).
$\cos(\theta) = \frac{adjacent}{hypotenuse}$

STEP 9

Now, plug in the given values for $\theta$ and the adjacent side to calculate the length of the ladder.
$\cos(60^{\circ}) = \frac{2 \mathrm{~m}}{length}$

STEP 10

Rearrange the equation to solve for the length.
$length = \frac{2 \mathrm{~m}}{\cos(60^{\circ})}$

STEP 11

Substitute the given value for $\cos(60^{\circ})$ into the equation.
$length = \frac{ \mathrm{~m}}{0.5}$

STEP 12

Calculate the length of the ladder.
$length = \frac{2 \mathrm{~m}}{0.5} =4 \mathrm{~m}$ So, the height of the wall is $.464 \mathrm{~m}$ and the length of the ladder is $4 \mathrm{~m}$ .

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QuestionA ladder leans against a wall at 60∘60^{\circ}60∘. The base is 2 m2 \mathrm{~m}2 m from the wall. Find the wall height and ladder length. [sin⁡60=0.866,cos⁡60=0.5,tan⁡60=1.732][\sin 60=0.866, \cos 60=0.5, \tan 60=1.732][sin60=0.866,cos60=0.5,tan60=1.732]

Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Now, plug in the given values for θ\thetaθ and the adjacent side to calculate the height of the wall.tan⁡(60∘)=height2 m\tan(60^{\circ}) = \frac{height}{2 \mathrm{~m}}tan(60∘)=2 mheight​

STEP 5

Rearrange the equation to solve for the height.height=tan⁡(60∘)×2 mheight = \tan(60^{\circ}) \times2 \mathrm{~m}height=tan(60∘)×2 m

STEP 6

Substitute the given value for tan⁡(60∘)\tan(60^{\circ})tan(60∘) into the equation.height=1.732×2 mheight =1.732 \times2 \mathrm{~m}height=1.732×2 m

STEP 7

Calculate the height of the wall.height=1.732×2 m=3.464 mheight =1.732 \times2 \mathrm{~m} =3.464 \mathrm{~m}height=1.732×2 m=3.464 m

STEP 8

STEP 9

Now, plug in the given values for θ\thetaθ and the adjacent side to calculate the length of the ladder.cos⁡(60∘)=2 mlength\cos(60^{\circ}) = \frac{2 \mathrm{~m}}{length}cos(60∘)=length2 m​

STEP 10

Rearrange the equation to solve for the length.length=2 mcos⁡(60∘)length = \frac{2 \mathrm{~m}}{\cos(60^{\circ})}length=cos(60∘)2 m​

STEP 11

Substitute the given value for cos⁡(60∘)\cos(60^{\circ})cos(60∘) into the equation.length= m0.5length = \frac{ \mathrm{~m}}{0.5}length=0.5 m​

STEP 12

Calculate the length of the ladder.length=2 m0.5=4 mlength = \frac{2 \mathrm{~m}}{0.5} =4 \mathrm{~m}length=0.52 m​=4 mSo, the height of the wall is .464 m.464 \mathrm{~m}.464 m and the length of the ladder is 4 m4 \mathrm{~m}4 m.

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QuestionA ladder leans against a wall at $60^{\circ}$ . The base is $2 \mathrm{~m}$ from the wall. Find the wall height and ladder length. $[\sin 60=0.866, \cos 60=0.5, \tan 60=1.732]$

Now, plug in the given values for $\theta$ and the adjacent side to calculate the height of the wall.
$\tan(60^{\circ}) = \frac{height}{2 \mathrm{~m}}$

Rearrange the equation to solve for the height.
$height = \tan(60^{\circ}) \times2 \mathrm{~m}$

Substitute the given value for $\tan(60^{\circ})$ into the equation.
$height =1.732 \times2 \mathrm{~m}$

Calculate the height of the wall.
$height =1.732 \times2 \mathrm{~m} =3.464 \mathrm{~m}$

Now, plug in the given values for $\theta$ and the adjacent side to calculate the length of the ladder.
$\cos(60^{\circ}) = \frac{2 \mathrm{~m}}{length}$

Rearrange the equation to solve for the length.
$length = \frac{2 \mathrm{~m}}{\cos(60^{\circ})}$

Substitute the given value for $\cos(60^{\circ})$ into the equation.
$length = \frac{ \mathrm{~m}}{0.5}$

Calculate the length of the ladder.
$length = \frac{2 \mathrm{~m}}{0.5} =4 \mathrm{~m}$ So, the height of the wall is $.464 \mathrm{~m}$ and the length of the ladder is $4 \mathrm{~m}$ .