According to a certain estimate, the depth N(t), in centimeters, of

the expression is N(t)= -20(t-5)^²+500
(of course valid after 2:00 in the morning)

"the depth would be maximum" means the value of the above expression should be maximum
or the value of square term (which has a negative 20 attached to it) should be minimum i.e. zero

the square part is zero at t=5

so the time at which the depth is maximum is 2:00 + 5 hrs= 7:00 (B)

if -20(t-5)^²=0
then t = 5
But Why is the 500 of the equation is not considered? Show your complete calculation.

A similar problem: https://gmatclub.com/forum/an-object-thrown-directly-upward-88791.html

Hi All,

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:

GMAT assassins aren't born, they're made,
Rich

Since -20(t - 5)^2, will be a nonpositive number, its maximum value is 0 when t = 5, and the maximum value of the function will then be:

N(5) = -20(5 - 5)^2 + 500 = 500

Thus, the maximum depth is at 2am + 5 hours = 7am.

Answer: B

Can someone explain why / how you would know not to simplify the original provided equation (-20(t-5)^2 + 500)? When you simplify the equation, you seems to get a different answer. Please see work below:

-20(t-5)^2 + 500
-20(t^2 - 10t + 25) + 500 <-- foil
-20t^2 + 200t - 500 + 500 <-- distributed the -20
-20t^2 + 200t
20t(t + 10t) <-- simplified formula

Given the 20t(t+10t) is simplified correctly, then the water level will continue to grow with every hour with the latest hour being the maximum. This is different than the answer provided, where t is at the max once 5 hours past.

Hi rjacobson99,

In the last "step" of you work, you have not properly accounted for the 'minus' sign. If you want to factor out "20t", then that's fine, but here's what you would be left with:

-20t^2 + 200t
(20t)(-t + 10)
(20t)(10 - t)

The maximum result will occur when t = 5.

GMAT assassins aren't born, they're made,
Rich

Hi Rich,

In the correct version of the formula (20t)(10-t), is there a way to determine the inflection point of t = 5 without having to plug in values from t = 1 to 6? I.e. is there a shortcut?

RJ

Hi rjacobson99,

Unfortunately, manipulating the equation in the way that you did places the variable "t" in both pairs of parentheses, so there isn't an 'obvious' solution to maximize the value. However, in the original equation, there IS an obvious Number Property that you can use...

N(t) = -20(t - 5)^2 + 500 for 0 ≤ t ≤ 10.

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 negative term - 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

GMAT assassins aren't born, they're made,
Rich

N(t)= -20\((t - 5)^2\) + 500

For this to be maximum, we need -20\((t - 5)^2\) minimum (zero) and that will happen when t = 5.

Hence, after 5 hours from 2:00 a.m. i.e., 7 a.m.

Answer B

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Answer: Option B

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I actually didn't use the "logic" approach, but rather the mathematics approach.
So for all those who are familiar with deriving, it can be solved in 3 steps.

N=-20(t-5)²+500 it is a maximum problem, so deriving this becomes to:

0=-40(t-5) and now simply work out t.

0=-40t + 200 add 40
40t = 200 divide by 40
t = 5

or

0 =-40(t-5) divide by 40
0 = t - 5 add 5
5 = t

plus the 2 hours from the question: 5 + 2 = 7

Calling all engineers!
Just differentiate the damn thing.

Posted from my mobile device

hey just wondering , how do we know that N ( t ) needs to be solved or not . i was just doing the same thing as u guys were doing but i was also substituting N ( t ) value .

 

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Check the highlighted parts in the stem. N(t) represents the depth, while we are asked to find such t for which the depth, \(N(t) = -20(t - 5)^2 + 500\), reaches its maximum.