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Intern  Joined: 12 May 2012
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A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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Question Stats: 80% (01:47) correct 20% (02:12) wrong based on 1881 sessions

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A ladder of a fire truck is elevated to an angle of 60° 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) $$35\sqrt{3}$$
(D) $$7 + 35\sqrt{3}$$
(E) $$7 + 42\sqrt{3}$$

Originally posted by sarb on 07 Jun 2012, 01:16.
Last edited by Bunuel on 07 Dec 2012, 05:04, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58454
A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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12
8
Notice that trigonometry is not tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many meet above the ground does the ladder reach?

A. 35
B. 42
C. 35 root 3
D. 7 + 35 root 3
E. 7 + 42 root 3

Look at the diagram below: Triangle ABC is a 30°-60°-90° triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio $$1 : \sqrt{3}: 2$$, the leg opposite 30° (AC) corresponds with $$1$$, the leg opposite 60° (BC) corresponds with $$\sqrt{3}$$ and the hypotenuse AB corresponds with 2. So, $$\frac{BC}{AB}=\frac{\sqrt{3}}{2}$$ --> $$\frac{BC}{70}=\frac{\sqrt{3}}{2}$$ --> $$BC=35\sqrt{3}$$.

Hence the leader reaches $$7+35\sqrt{3}$$ above the ground.

For more check Triangles chapter of Math Book: http://gmatclub.com/forum/math-triangles-87197.html

Hope it helps.

Attachment: Ladder.png [ 5.44 KiB | Viewed 62655 times ]

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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° 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) $$35\sqrt{3}$$
(D) $$7 + 35\sqrt{3}$$
(E) $$7 + 42\sqrt{3}$$

We are given that the ladder of a fire truck is elevated to an angle 60 degrees above the ground and that the ladder has a length of 70 feet. We are also given that the ladder is 7 feet above the ground. The best thing to do in this situation is to draw a diagram. Notice that the resulting triangle in the sketch is a 30-60-90 right triangle. Based on the given info, we don’t know that the ladder is leaning against a building whose side is perpendicular to the ground. The ratio of the sides of a 30-60-90 right triangle is x : x√3 : 2x. We see that the hypotenuse length of 70 feet is equal to the "2x" from the 30-60-90 ratio. Thus, we can set up an equation and solve for x.

70 = 2x

x = 35

Because x = 35, we know that the side opposite the 60-degree angle or, in this case, the height of the ladder, is 35√3. The height of the ladder is 35√3 feet and the base of the ladder is 7 feet above the ground; thus, we know that the ladder reaches a total height above the ground of 35√3 + 7 feet.

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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° 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) $$35\sqrt{3}$$
(D) $$7 + 35\sqrt{3}$$
(E) $$7 + 42\sqrt{3}$$

Here's an idea of what's going on.... 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 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... So, the TOTAL distance from the top of the ladder to the ground = 35√3 + 7

Cheers,
Brent
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Posts: 58454
Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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sv3n wrote:
I do not get the ratio 1 : 2 : √3 For me it is 1 : 2 : 3 (30° : 60° : 90°)... can anyone help?

Check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.
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Re: maths problem  [#permalink]

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Vertical height of ladder = 70.sin(60) = 35root3

Total height = 7 + 35root3
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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1
Since the questions says elevated at an angle of 60 deg, the base angle is 60 deg. By default, elevation is from baseline and since the ladder is placed parallel to x axis, the elevation is also from x axis.

Hope that helps!

thangvietnam wrote:
which one of 2 angles is 60?

the question is not clear.
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A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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Bunuel wrote:
Notice that trigonometry is not tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many meet above the ground does the ladder reach?

A. 35
B. 42
C. 35 root 3
D. 7 + 35 root 3
E. 7 + 42 root 3

Look at the diagram below: Triangle ABC is a 30°-60°-90° triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio $$1 : \sqrt{3}: 2$$, the leg opposite 30° (AC) corresponds with $$1$$, the leg opposite 60° (BC) corresponds with $$\sqrt{3}$$ and the hypotenuse AC corresponds with 2. So, $$\frac{BC}{AB}=\frac{\sqrt{3}}{2}$$ --> $$\frac{BC}{70}=\frac{\sqrt{3}}{2}$$ --> $$BC=35\sqrt{3}$$.

Hence the leader reaches $$7+35\sqrt{3}$$ above the ground.

For more check Triangles chapter of Math Book: http://gmatclub.com/forum/math-triangles-87197.html

Hope it helps.

Bunuel hello isn't a hypotenuse the longest one AB? why are you saying it is AC ? please explain have a great day ! D.
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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You have to assume the ladder is attached to the truck! That threw me off...
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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Bunuel wrote:
Notice that trigonometry is not tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many meet above the ground does the ladder reach?

A. 35
B. 42
C. 35 root 3
D. 7 + 35 root 3
E. 7 + 42 root 3

Look at the diagram below:
Attachment:
Triangle ABC is a 30°-60°-90° triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio $$1 : \sqrt{3}: 2$$, the leg opposite 30° (AC) corresponds with $$1$$, the leg opposite 60° (BC) corresponds with $$\sqrt{3}$$ and the hypotenuse AC corresponds with 2. So, $$\frac{BC}{AB}=\frac{\sqrt{3}}{2}$$ --> $$\frac{BC}{70}=\frac{\sqrt{3}}{2}$$ --> $$BC=35\sqrt{3}$$.

Hence the leader reaches $$7+35\sqrt{3}$$ above the ground.

For more check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.

Thank you that made it very simple Director  S
Joined: 09 Jun 2010
Posts: 713
Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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which one of 2 angles is 60?

the question is not clear.
Intern  Joined: 18 May 2013
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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I do not get the ratio 1 : 2 : √3 For me it is 1 : 2 : 3 (30° : 60° : 90°)... can anyone help?

Intern  Joined: 18 May 2013
Posts: 7
Location: Germany
GMAT Date: 09-27-2013
Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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Bunuel wrote:
sv3n wrote:
I do not get the ratio 1 : 2 : √3 For me it is 1 : 2 : 3 (30° : 60° : 90°)... can anyone help?

Check Triangles chapter of Math Book: ...

Hope it helps.

OK.. IT´S GIVEN. Thanks a lot! Intern  Joined: 03 Aug 2013
Posts: 1
Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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I understand the ratio x:x(sqrt)3:2x and how to get 7+35sqrt3, but how is this not a 3-4-5 right triangle with lengths of 42-56-70? Thank you for the added explanation.
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Posts: 58454
Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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bgourlay13 wrote:
I understand the ratio x:x(sqrt)3:2x and how to get 7+35sqrt3, but how is this not a 3-4-5 right triangle with lengths of 42-56-70? Thank you for the added explanation.

If it's 42-56-70, what is x then? Also, we get that BC, the smallest side, is $$35\sqrt{3}$$ not 42.

Hope it's clear.
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° 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) $$35\sqrt{3}$$
(D) $$7 + 35\sqrt{3}$$
(E) $$7 + 42\sqrt{3}$$

sin x = opp side/hypotenuse side

sin 60 =root(3)/2= x/70

x= 35 * root(3)

total height = 35 root (3)+ 7     Intern  Joined: 17 May 2012
Posts: 34
A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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Hi All,

Another approach to solve this question is if you know the values of sin/cos angles. In this case sin 60 = Perpendicular/Hypotenuse i.e. sin 60 = BC/AB.
As sin 60 =$$\sqrt{3}/2$$, we get $$\sqrt{3}/2$$ = BC/70, thus BC = 35$$\sqrt{3}$$

From that you can get total height from the base 7 + 35$$\sqrt{3}$$ as the final answer. Hope that helps.
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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Bunuel wrote:
Notice that trigonometry is not tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many meet above the ground does the ladder reach?

A. 35
B. 42
C. 35 root 3
D. 7 + 35 root 3
E. 7 + 42 root 3

Look at the diagram below:
Attachment:
Triangle ABC is a 30°-60°-90° triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio $$1 : \sqrt{3}: 2$$, the leg opposite 30° (AC) corresponds with $$1$$, the leg opposite 60° (BC) corresponds with $$\sqrt{3}$$ and the hypotenuse AC corresponds with 2. So, $$\frac{BC}{AB}=\frac{\sqrt{3}}{2}$$ --> $$\frac{BC}{70}=\frac{\sqrt{3}}{2}$$ --> $$BC=35\sqrt{3}$$.

Hence the leader reaches $$7+35\sqrt{3}$$ above the ground.

--------------------------------------------------------------------------------------------------------------

I understand this but I drew it sideways and got the answer of (B) 42

I'm not sure how you determined which side is the truck and which is the wall. Couldn't it have been a very long truck and a short wall that the ladder was against?
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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dave13 wrote:
Bunuel wrote:
Notice that trigonometry is not tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

sarb wrote:
A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many meet above the ground does the ladder reach?

A. 35
B. 42
C. 35 root 3
D. 7 + 35 root 3
E. 7 + 42 root 3

Look at the diagram below: Triangle ABC is a 30°-60°-90° triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio $$1 : \sqrt{3}: 2$$, the leg opposite 30° (AC) corresponds with $$1$$, the leg opposite 60° (BC) corresponds with $$\sqrt{3}$$ and the hypotenuse AC corresponds with 2. So, $$\frac{BC}{AB}=\frac{\sqrt{3}}{2}$$ --> $$\frac{BC}{70}=\frac{\sqrt{3}}{2}$$ --> $$BC=35\sqrt{3}$$.

Hence the leader reaches $$7+35\sqrt{3}$$ above the ground.

For more check Triangles chapter of Math Book: http://gmatclub.com/forum/math-triangles-87197.html

Hope it helps.

Bunuel hello isn't a hypotenuse the longest one AB? why are you saying it is AC ? please explain have a great day ! D.

There was a typo: AC instead of AB. Edited. Thank you.
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Re: A ladder of a fire truck is elevated to an angle of 60° and  [#permalink]

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How do i know this is a 60-30-90 triangle and not 60-60-60?

Posted from my mobile device Re: A ladder of a fire truck is elevated to an angle of 60° and   [#permalink] 23 Mar 2019, 21:52

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