Last Updated on May 10, 2023
GMAT OFFICIAL GUIDE PS
Solution:
We are given that at t hours past 2:00 a.m., the depth of the water in a tank is determined by the formula N(t)= -20(t-5)^2 + 500 for 0 ≤ t ≤ 10. We must determine which value of “t” provides the maximum depth. Using the order of operations, we follow four steps.
Step 1: subtract 5 from t
Step 2: square the result of t – 5
Step 3: multiply the square of (t – 5) by -20
Step 4: add 500 to the product of the square of (t – 5) and -20
If we focus on step 3, we see that we are multiplying a perfect square by a negative number; thus, the product will be non-positive. Since the last step is then adding that to a positive 500, our goal should be to get -20(t – 5)^2 as close to zero as possible. Doing so will provide us with the maximum possible depth.
The value for t that gets -20(t – 5)^2 as close to zero as possible is t = 5. Actually the value of -20(t – 5)^2 will be exactly 0 when t = 5 and we will have:
N(5)= -20(5 – 5)^2 500
N(5)= -20(0)^2 +500
N(5)= 500
500 centimeters is the maximum depth. This would occur 5 hours after 2 am, so at 7 am the depth is at its maximum.
Alternate solution:
If you have a strong algebra background, you will recognize that the graph of the function N(t)= -20(t-5)^2 + 500 is a downward parabola whose axis of symmetry has been shifted 5 units to the right of the origin. Thus, you would know that the vertex of the parabola (whose N(t) value is the maximum value of the water height) would have a t-coordinate of t = 5 hours. Add 5 hours to 2 am, and you get the same answer of 7 am.
Answer: B
