Given that \(h=150-16(t-3)^2\).

In order to maximize \(h\), we need to minimize \(16(t-3)^2\) (since it's subtracted from 150), which means minimizing \((t-3)^2\). Now, since \((t-3)^2\) is always non-negative, then the smallest possible value of \((t-3)^2\) is 0, for \(t=3\).

Two seconds later or for \(t=3+2=5\), the height will be \(h=150-16(5-3)^2=86\).

Answer: B.

Hope it's clear.
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max height at t=3, 150

2 sec after t=3 means t=5.

substituting h= -64 + 150 = 86

Method 1: -16(t-3)^2 would reduce the value of 150 and h would be maximum when -16(t-3) ^2 = 0 ie t = 3
Method 2: using derivatives dh/dt = 0.

First find out what height would be considered as 'maximum' given the equation:

Since 150 is a constant, find out what the product's maximum value can be. It is a product of a negative number and a squared parenthesis; this product would ALWAYS be negative UNLESS the parenthesis yields zero - this would be the point at which the product would have maximum value.

For t = 3, the expression would yield 0 + 150. Hence, the maximum possible height given by this equation would be 150 feet. Therefore, after reaching this height, the object would start falling. The 'start' time of falling would be after 3 seconds, so adding 2 seconds would be 5:

h = -16 x (5-3)^2 + 150
h = -16 x 4 + 150
h = 86

Attached is a visual that should help.

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We are given an equation h = –16(t – 3)^2 + 150, with the following information:

h = height of h feet

t = number of seconds

We need to determine the height, in feet, 2 seconds after it reaches maximum height. So we first need to determine the value of t when the object's height h is the maximum. In other words, we need to determine the maximum value for this equation.

We first focus on “-16(t - 3)^2”. We ascertain that (t – 3)^2 is always either positive or 0. However, when (t – 3)^2 is multiplied by –16, a negative number, the product will be negative. Thus, the best we can do is to have the expression -16(t - 3)^2 equal 0, which would yield the maximum value of –16(t – 3)^2 + 150. We can obtain this value by letting t = 3.

We now know that the object reaches its maximum height at t = 3 (and the maximum height is 150 ft). However, we want the height of the object 2 seconds after it reaches the maximum height. Thus, we want the height at t = 5 since 3 + 2 = 5. Thus, we can plug in 5 for t and solve for h.

h = -16(t - 3)^2 + 150

h = -16(5 - 3)^2 + 150

h = -16(2)^2 + 150

h = -16 x 4 + 150

h = -64 + 150

h = 86

Answer B
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Hi abhibad,

The equation that the prompt gives us to work with tells us the height of the ball after T seconds have elapsed.

So, after 3 seconds (meaning T=3), the height of the ball is 150 feet.

Two seconds AFTER that would be the 5 second 'mark' (meaning T=5). In your calculation, you plugged in T=2.

GMAT assassins aren't born, they're made,
Rich
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The formula h = -16 (t - 3)² + 150 allows us to determine the height of the object at any time. For what value of t is -16(t-3)² + 150 MAXIMIZED(in other words, the object is at its maximum height)?

It might be easier to answer this question if we rewrite the formula as h = 150 - 16(t-3)²
To MAXIMIZE the value of h, we need to MINIMIZE the value of 16(t-3)² and this means minimizing the value of (t-3)²
As you can see,(t-3)² is minimized when t = 3.

We want to know the height 2 seconds AFTER the object's height is maximized, so we want to know that height at 5 seconds (3+2)

At t = 5, the height = 150 - 16(5 - 3)²
= 150 - 16(2)²
= 150 - 64
= 86
Answer:

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Hi santro789,

This is an example of a 'limit' question - and this particular 'design' is relatively rare (you might not see it at all on Test Day). In the broader sense though, the concept isn't that difficult. You're told that there's a MAXIMUM height, so you have to think about what has to happen to achieve the highest possible height given that equation (and when you think of everything in those terms, you're really just dealing with a Number Property and using critical thinking skills - NOT 'math' skills). Even if you don't spot the pattern here, you can still 'brute force' the solution: just plug in increasing values of T until you get the answer.

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