Question

HELP NOW! find x

Explanation please too
My teacher said answer is under 100 and over 20 and it is a decimal so please round to the nearest tenth

Answer 1

97.0 ft

Step-by-step explanation:

The given diagram shows two right triangles, both of which share a common height (x).

To find the height of the lighthouse (x), we can create two equations by substituting the information for each triangle into the tangent trigonometric ratio.

`\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=(O)/(A)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$O$ is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\end{array}}`

Triangle 1

• Angle = 30°
• x is the side opposite the angle.
• y is the side adjacent the angle.

Therefore, substituting these values into the tangent ratio gives:

`\tan 30^(\circ)=(x)/(y)`

Rearrange to isolate y:

`y \cdot \tan 30^(\circ)=y \cdot (x)/(y)`

`y \tan 30^(\circ)=x`

`y=(x)/(\tan 30^(\circ))`

Triangle 2

• Angle = 25°
• x is the side opposite the angle.
• (y + 40) is the side adjacent the angle.

Therefore, substituting these values into the tangent ratio gives:

`\tan 25^(\circ)=(x)/(y+40)`

Rearrange to isolate y:

`(y+40)\cdot \tan 25^(\circ)=(y+40)\cdot (x)/(y+40)`

`y\tan 25^(\circ)+40 \tan 25^(\circ)=x`

`y\tan 25^(\circ)=x-40 \tan 25^(\circ)`

`y=(x-40 \tan 25^(\circ))/(\tan 25^(\circ))`

Now, we have created two equations:

`\begin{cases}y=(x)/(\tan 30^(\circ))\\\\y=(x-40 \tan 25^(\circ))/(\tan 25^(\circ))\end{cases}`

Substitute the first equation into the second equation, so that we have an equation in terms of x only:

`(x)/(\tan 30^(\circ))=(x-40 \tan 25^(\circ))/(\tan 25^(\circ))`

Now, solve for x:

`(x)/(\tan 30^(\circ))=(x)/(\tan 25^(\circ))-(40 \tan 25^(\circ))/(\tan 25^(\circ))`

`(x)/(\tan 30^(\circ))=(x)/(\tan 25^(\circ))-40`

`(x)/(\tan 30^(\circ))-(x)/(\tan 25^(\circ))=-40`

`(x\tan 25^(\circ))/(\tan 30^(\circ)\tan 25^(\circ))-(x\tan 30^(\circ))/(\tan 30^(\circ)\tan 25^(\circ))=-40`

`(x\tan 25^(\circ)-x\tan 30^(\circ))/(\tan 30^(\circ)\tan 25^(\circ))=-40`

`x\tan 25^(\circ)-x\tan 30^(\circ)=-40\tan 30^(\circ)\tan 25^(\circ)`

`x(\tan 25^(\circ)-\tan 30^(\circ))=-40\tan 30^(\circ)\tan 25^(\circ)`

`x=(-40\tan 30^(\circ)\tan 25^(\circ))/(\tan 25^(\circ)-\tan 30^(\circ))`

`x=96.9800149...`

`x=97.0\; \sf ft`

Therefore, the height of the lighthouse (x) is 97.0 ft (rounded to the nearest tenth).

Answer 2

x = 97.0

y = 168.0

Step-by-step explanation:

Make equations using trigonometric ratios:

• 
  `(x)/(y) =\tan(30)` _____(1)

• 
  `(x)/(y+40)=tan(25)` ______(2)

Make x the subject for both:

• 
  `x =y\tan(30)` ____(1)

• 
  `x=(y+40)tan(25)` ____(2)

Solve them simultaneously:

`ytan(30)=(y+40)tan(25)`

`ytan(30)=ytan(25)+40tan(25)`

`ytan(30)-ytan(25)= 40tan(25)`

`y(tan(30)-tan(25))= 40tan(25)`

`y= 40tan(25)/(tan(30)-tan(25)) = 167.97 = 168.0`

Then, find value of x

`x = ytan30 = 167.97tan(30) = 96.98 = 97.0`