According to a certain estimate, the depth N(t), in cm, of water in a tank at t hours past 2 a.m. is given by N(t)=-20(t-5)^2+500 for all t from 0 to 10 hours, inclusive. According to this estimate, at what time does the tank's depth reach its maximum?
A) 5:30
B) 7:00
C) 7:30
D) 8:00
E) 9:00
My dilemma:
1) I understand that the equation is meant to equal the maximum depth at a certain value of t. How does one test values of t from 0 to 10 in less than 2 minutes?? Is there a quicker way to solve this?
2) If the answer to 1) is "when t-5=0" then how does one conclude that 500 is the target depth? For example, why could it not be 420 when t=7?
3)Is there something inherent in these types of questions that signals a given term in the formula (in this case, the +500) is the target value, even when the question doesn't specifically say so?
-L
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Depth = 500 - 20(t-5)².According to a certain estimate, the depth N(t), in centimeters, of the water in a certain tank at t hours past 2:00 in the morning is given by N(t)= -20(t-5)²+500 for 0≤t≤10. According to this estimate, at what time in the morning does the depth of the water in the tank reach its maximum?
a) 5:30 b) 7:00 c) 7:30 d) 8:00 e)9:00
To MAXIMIZE the depth, we must MINIMIZE the value in red.
Since the square of a value cannot be negative, (t-5)² ≥ 0.
Thus, the value in red will be minimized when (t-5)² = 0.
Since (t-5)² = 0 when t=5, the depth will be at its maximum 5 hours after 2am:
2am + 5 hours = 7am.
The correct answer is B.
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Hi lfa713,
These specific types of "limit" questions are relatively rare on Test Day, although you'll likely be tested on the concept at least once. Whenever you're asked to minimize or maximize a value, you should look to do something with the other "pieces" of the equation (usually involving maximizing or minimizing those pieces).
In the given equation, notice how you have two "parts": the -20(something) and a +500. Here, to MAXIMIZE the value of N(t), we have to minimize the "impact" that the -20(something) has on the +500. By making that first part equal 0, we'll be left with 0 + 500. Mathematically, we have to make whatever is inside the parentheses equal 0....
(T-5) = 0
T = 5
Since T represents the number of hours past 2:00am, we know that at 7:00am, the water will reach 500cm (the maximum value).
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
These specific types of "limit" questions are relatively rare on Test Day, although you'll likely be tested on the concept at least once. Whenever you're asked to minimize or maximize a value, you should look to do something with the other "pieces" of the equation (usually involving maximizing or minimizing those pieces).
In the given equation, notice how you have two "parts": the -20(something) and a +500. Here, to MAXIMIZE the value of N(t), we have to minimize the "impact" that the -20(something) has on the +500. By making that first part equal 0, we'll be left with 0 + 500. Mathematically, we have to make whatever is inside the parentheses equal 0....
(T-5) = 0
T = 5
Since T represents the number of hours past 2:00am, we know that at 7:00am, the water will reach 500cm (the maximum value).
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
