A Ladder of a Fire Truck is Elevated to an Angle of  \(60^\circ\) GMAT Problem Solving

Question: A ladder of a fire truck is elevated to an angle of \(60^\circ\)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?

1. 35
2. 42
3. \(35\sqrt3\)
4. \(7+35\sqrt3\)
5. \(7+42\sqrt3\)

Answer:
Solution with Explanation:

Approach Solution (1):

Look at the diagram below:

Triangle ABC is \(30^\circ-60^\circ-90^\circ\)triangle. Now, in a right angled triangle where the angles are \(30^\circ,60^\circ, and 90^\circ\) the sides are always in the ratio 1 :\(\sqrt3\): 2, the leg opposite\(30^\circ\)(AC) corresponds with 1, the leg opposite \(60^\circ\)(BC) corresponds with \(\sqrt3\) and the hypotenuse AB corresponds with 2.

So, \(\frac{BC}{AB}=\frac{\sqrt{3}}{2}\rightarrow\frac{BC}{70}=\frac{\sqrt{3}}{2}\rightarrow BC=35\sqrt3\)

Hence the ladder reaches \(7+35\sqrt3\) above the ground

Correct Option: D

Approach Solution (2):

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 that the big triangle is 35 times the size of the base triangle

This means that, on the big triangle, the side opposite the 60 degrees angle must be \(35\sqrt3\)

Now before we choose any option, 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 = \(7+35\sqrt3\)

Correct Option: D

Approach Solution (3):

We are given that the ladder of a fine truck is elevated to an angle 60 degrees above the ground and that the ladder has 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 x : \(x\sqrt3\) : 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 it for the value of 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 \(35\sqrt3\). The height of the ladder is \(35\sqrt3\) 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 \(35\sqrt3 + 7\) feet

Correct Option: D

“A ladder of a fire truck is elevated to an angle of \(60^\circ\) 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?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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