Trigonometry...height of a mountain. correct answer?

[dear_miss_katie]

Hello! I worked out this word problem in trig, but I have no idea if it is correct. I was wondering if someone would be so kind as to check my work, and perhaps steer me in the right direction. I would really appreciate it.

While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the pak is 3.5 degrees. After you drive 14 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain.

I drew a picture to show the relationship between triangles, and used the trig ratios tan 81degrees= y/x and tan 86.5 degrees= y+13/ x. Our teacher told us to isolate the "y"s and so I eventually was able to use substitution and get the answer of 3.44 miles. Is this anywhere near the right answer, or have I messed it up? Thank you so much for your time!

 

[galactus]

I don't believe that is correct. You are right about the substitution method.

You can create two equations and solve.

Let y=height of the mountain and 14+x the distance from the first point with the 3 degree angle,

you have $\displaystyle \frac{y}{14+x}=tan(\frac{7}{2})$

After you drive closer by 14 miles:

$\displaystyle y=xtan(9)$

$\displaystyle \frac{x\cdot{tan(9)}}{14+x}=tan(\frac{7}{2})$

Can you solve for x?.

 

[dear_miss_katie]

Hi again! Thank you so much for your very helpful reply! I realize that I have made a typing error...the question states "after you drive 13 miles closer" instead of fourteen. The technique is the same, though. I tried to solve for x, and got some nasty decimals..I suppose that's normal. I ended up with 8.18 miles this time. Thank you so much for helping me

 

[galactus]

Yes, correct. But remember, that is the distance from the mountain, not the height. y is the height. Sub your x=8.18 into y=xtan(9) to find the mountain height.

 

[dear_miss_katie]

ohhhh! I forgot about that! thanks!