Bunuel wrote:
The maximum height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?
A. 12 feet per second
B. 16 feet per second
C. 18 feet per second
D. 24 feet per second
E. 48 feet per second
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:Typically with direct proportion problems, you will be given “before” and “after” values. Simply set up ratios to solve the problem. For example, \(\frac{y_1}{x_1}\) can be used for the “before” values and , \(\frac{y_2}{x_2}\) can be used for the “after” values. You then write, \(\frac{y_1}{x_1}=\frac{y_2}{x_2}\), since both ratios are equal to the same constant k. Finally, you solve for the unknowns.
In the problem given above, be sure to note that the direct proportion is between the height and the square of the velocity, not the velocity itself. Therefore, write the proportion as \(\frac{h_1}{(v_1)^2}=\frac{h_2}{(v_2)^2}\). Substitute the known values h1 = 4, v1 = 16, and h2 = 9:
\(\frac{4}{(16)^2}=\frac{9}{(v_2)^2}\)
\(v_2=24\).
The object must be thrown upward at 24 feet per second.
Answer: D.
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