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Geometry Triangle Problem

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by me_1234 » Wed Dec 10, 2014 6:21 pm
A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach?

A/ 35
B/ 42
C/ 35√3
D/ 7 + 35√3 (Correct Answer)
E/ 7 + 42√3

When I did this problem I did the following - can someone tell me where I went wrong?

Using the properties of a 30-60-90 triangle, I set up a ratio:

70 = 2
x √3

70 * √3 = 2x, then divide by 2:

35 * (√3/2) = x

So my answer was 7 + 35* (√3/2)
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by Brent@GMATPrepNow » Wed Dec 10, 2014 7:26 pm
melanie.espeland wrote:A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach?

A/ 35
B/ 42
C/ 35√3
D/ 7 + 35√3 (Correct Answer)
E/ 7 + 42√3

When I did this problem I did the following - can someone tell me where I went wrong?

Using the properties of a 30-60-90 triangle, I set up a ratio:

70 = 2
x √3

70 * √3 = 2x, then divide by 2:

35 * (√3/2) = x

So my answer was 7 + 35* (√3/2)
You ALMOST had it.
Let's take it from:
70√3 = 2x
Divide both sides by 2 to get: 35√3 = x [not x = 35(√3/2)]

Cheers,
Brent
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by me_1234 » Wed Dec 10, 2014 8:29 pm
Hi Brent,

When you have 70 * √3 = 2x, then divide by 2, why isn't it 70/2 and √3/2? It looks like you are only dividing 70 by 2.

Thanks

Melanie
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by Brent@GMATPrepNow » Wed Dec 10, 2014 8:44 pm
melanie.espeland wrote:Hi Brent,

When you have 70 * √3 = 2x, then divide by 2, why isn't it 70/2 and √3/2? It looks like you are only dividing 70 by 2.

Thanks

Melanie
Let's examine an easier question:
How about (10)(4) = 2x
Would we divide the 10 AND the 4 by 2? No.
We take (10)(4) and divide by 2 to get: (10)(4)/2 = (2)(5)(4)/2 = (5)(4) = 20

Likewise, if 70√3 = 2x, then x = (70√3)/2 = (2)(35)(√3)/2 = (35)(√3)

Cheers,
Brent






Cheers,
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by MartyMurray » Wed Dec 10, 2014 9:21 pm
melanie.espeland wrote:Hi Brent,

When you have 70 * √3 = 2x, then divide by 2, why isn't it 70/2 and √3/2? It looks like you are only dividing 70 by 2.

Thanks

Melanie
That's right, Melanie. One does only divide the 70, or alternatively you divide the entire thing, (70 * √3), by 2. What you are doing is actually dividing twice. If the 70 * √3 were instead 70 + √3, THEN you would need to divide each of them, but since they are multiplied, you need to divide only once.

Here's another way to look at it.

The proportions of the sides of a 30 - 60 - 90 triangle are 1 : √3 : 2.

The hypotenuse is the 2, and in this case the hypotenuse has length 70. So to work with this triangle, let's divide the 70 by 2 to get our basic unit for the lengths of the sides, 70/2 = 35.

Now to get the lengths of the three sides, we can plug the 35 into the ratio to get 1(35) : √3(35) : 2(35).

The height above the ground is 7 + the length of that √3(35) side. So it's 7 + √3(35), or as they put it 7 + 35√3.
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by me_1234 » Thu Dec 11, 2014 9:30 am
Thank you Brent and Marty, totally get it now.
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by Mathsbuddy » Fri Dec 19, 2014 6:25 am
Height, h = 7 + 70sin60(degrees)
but sin60 = SQRT[3/4]= SQRT[3]/2
So, h = 7 + 70 SQRT[3]/2 = 7 + 35 SQRT[3]
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by [email protected] » Fri Dec 19, 2014 8:19 am
Hi Mathsbuddy,

While using trigonometry to answer this question is one way to get to the solution, it's worth noting that NO GMAT questions will ever require you to use trigonometry to get to the solution.

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by Brent@GMATPrepNow » Sat Oct 26, 2019 9:13 am
me_1234 wrote:A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach?

A/ 35
B/ 42
C/ 35√3
D/ 7 + 35√3 (Correct Answer)
E/ 7 + 42√3
Here's an idea of what's going on....
Image
When we compare the big triangle with the BASE 30-60-90 special triangle (which you must memorize for test day!), we can see that the hypotenuse of the big triangle is 35 times as long as the hypotenuse of the base triangle.
This means the big triangle is 35 times the size of the base triangle.

This means that, on the big triangle, the side opposite the 60 degree angle must be 35√3
Image

Now before we (incorrectly) choose answer choice C, we must keep in mind that the question tells us the base of the ladder is 7 feet above the ground...
Image

So, the TOTAL distance from the top of the ladder to the ground = 35√3 + 7

Answer: D

Cheers,
Brent
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