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Bad Calculations-No Calculus

by sanju09 » Thu Jan 23, 2014 11:02 pm
A rocket is fired into the air, and its height in meters at any given time t in seconds can be calculated using the formula h(t) = 1600 + 196t - 4.9t^2. What is the literal maximum height in meters achieved by the rocket?
A. 3560
B. 4200
C. 4560
D. 5100
E. 5560


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Last edited by sanju09 on Wed Jan 29, 2014 3:57 am, edited 1 time in total.
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by Brent@GMATPrepNow » Sun Jan 26, 2014 9:48 am
Hi Sanjeev,

Perhaps I'm missing something, but it appears that we can answer the target question without even examining the statements. I believe that the height formula [h(t) = 1600 + 196t - 4.9t²] provides all the information needed to determine the maximum height.

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Brent
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by sanju09 » Mon Jan 27, 2014 10:52 pm
Brent@GMATPrepNow wrote:Hi Sanjeev,

Perhaps I'm missing something, but it appears that we can answer the target question without even examining the statements. I believe that the height formula [h(t) = 1600 + 196t - 4.9t²] provides all the information needed to determine the maximum height.

Cheers,
Brent
Yeah, I know this :); but wondering how without "Differential Calculus"? Please enlighten. Regards
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by Brent@GMATPrepNow » Tue Jan 28, 2014 8:30 am
Okay, so it's not really a Data Sufficiency question then?
A rocket is fired into the air, and its height in meters at any given time t in seconds can be calculated using the formula h(t) = 1600 + 196t - 4.9t². What is the literal maximum height achieved by the rocket?
Aside: I think this question is out of scope for the GMAT

We can rewrite the formula using a technique called "Completing the Square"
Start with: 1600 + 196t - 4.9t²
Rearrange: -4.9t² + 196t + 1600
Factor (partially): -4.9(t² - 40t) + 1600

IMPORTANT: Here come the "completing the square" part. What must we add to t² - 40t to get a square? That is, we want to add something to t² - 40t so that we can factor it to get (t - something)². To find this mystery value, we take 40 and halve it to get 20, and then take 20 and square it to get 400. Notice that t² - 40t + 400 = (t - 20)² ...perfect!

So, we'll complete the square by adding 400 to the part in the brackets.
NOTE: We can't just add 400 (willy nilly) to the part in the brackets since this fundamentally changes the expression. So, if we ADD 400, we must also SUBTRACT 400. That way, we are simply adding ZERO, which changes nothing.

Okay, so we were here: -4.9(t² - 40t) + 1600
Add "zero" to the part in brackets: -4.9(t² - 40t + 400 - 400) + 1600
To remove the - 400 from the part in brackets, we'll multiply it by -4.9 to get...
-4.9(t² - 40t + 400) + 1960 + 1600
Simplify: -4.9(t² - 40t + 400) + 3560
Factor: -4.9(t - 20)² + 3560

So, we've successfully rewritten the height formula as h(t) = -4.9(t - 20)² + 3560
Here, we can see that the MAXIMUM value of this function will occur when we MINIMIZE the value of (t - 20)²
(t - 20)² is MINIMIZED with t = 20, which means the MAXIMUM height occurs when t = 20
Finally, to determine the maximum height, we'll plug 20 into the new height formula to get:
h(20) = -4.9(t - 20)² + 3560
= 3560

So, the maximum height is 3560 meters

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Wed Jan 29, 2014 4:02 am, edited 1 time in total.
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by sanju09 » Tue Jan 28, 2014 11:30 pm
Brent@GMATPrepNow wrote:Okay, so it's not really a Data Sufficiency question then?
A rocket is fired into the air, and its height in meters at any given time t in seconds can be calculated using the formula h(t) = 1600 + 196t - 4.9t². What is the literal maximum height achieved by the rocket?
Aside: I think this question is out of scope for the GMAT

We can rewrite the formula using a technique called "Completing the Square"
Start with: 1600 + 196t - 4.9t²
Rearrange: -4.9t² + 196t + 1600
Factor (partially): -4.9(t² - 20t) + 1600

IMPORTANT: Here come the "completing the square" part. What must we add to t² - 20t to get a square? That is, we want to add something to t² - 20t so that we can factor it to get (t - something)². To find this mystery value, we take 20 and halve it to get 10, and then take 10 and square it to get 100. Notice that t² - 20t + 100 = (t - 10)² ...perfect!

So, we'll complete the square by adding 100 to the part in the brackets.
NOTE: We can't just add 100 (willy nilly) to the part in the brackets since this fundamentally changes the expression. So, if we ADD 100, we must also SUBTRACT 100. That way, we are simply adding ZERO, which changes nothing.

Okay, so we were here: -4.9(t² - 20t) + 1600
Add "zero" to the part in brackets: -4.9(t² - 20t + 100 - 100) + 1600
To remove the - 100 from the part in brackets, we'll multiply it by -4.9 to get...
-4.9(t² - 20t + 100) - 490 + 1600
Simplify: -4.9(t² - 20t + 100) + 1110
Factor: -4.9(t - 10)² + 1110

So, we've successfully rewritten the height formula as h(t) = -4.9(t - 10)² + 1110
Here, we can see that the MAXIMUM value of this function will occur when we MINIMIZE the value of (t - 10)²
(t - 10)² is MINIMIZED with t = 10, which means the MAXIMUM height occurs when t = 10
Finally, to determine the maximum height, we'll plug 10 into the new height formula to get:
h(10) = -4.9(10 - 10)² + 1110
= 1110

So, the maximum height is 1110 meters

Cheers,
Brent
The discussion that follow has nothing to do with the GMAT, so please ignore.

Hi Brent,

Excellent! Thanks for reminding completion of square technique. It is always possible in a quadratic expression. I tried it using Differential Calculus approach:

Differentiate h(t) = 1600 + 196t - 4.9t^2 w.r.t. t:

d/dt h(t) = 196 - 9.8t.

For Maxima and Minima d/dt h(t) = 196 - 9.8t = 0, this happens at t = 20. We then test for Maxima or Minima by taking the second derivative value of h(t) at t = 20. If it's positive, the value represents a Minimum; and if it's negative, the value represents a Maximum.

d/dt[d/dt h(t)] = -9.8, negative, irrespective of the value of t, hence the height is maximum at t = 20, and the real maximum height

h(20) = 1600 + 196(20) - 4.9(20)^2

h(20) = 1600 + 3920 - 1960 = 3560 meters.

[spoiler]Why our answers different?[/spoiler]
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by navinCH » Wed Jan 29, 2014 3:50 am
The mistake in that work is 20 in place of 40. h(t)=1600+196t-4.9t^2=1600-4.9(t^2=40), rest is good.
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by sanju09 » Wed Jan 29, 2014 3:59 am
navinCH wrote:The mistake in that work is 20 in place of 40. h(t)=1600+196t-4.9t^2=1600-4.9(t^2=40), rest is good.
Oh yes, you are correct navinCH.
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by Brent@GMATPrepNow » Wed Jan 29, 2014 4:04 am
sanju09 wrote: The discussion that follow has nothing to do with the GMAT, so please ignore.

Hi Brent,

Excellent! Thanks for reminding completion of square technique. It is always possible in a quadratic expression. I tried it using Differential Calculus approach:

Differentiate h(t) = 1600 + 196t - 4.9t^2 w.r.t. t:

d/dt h(t) = 196 - 9.8t.

For Maxima and Minima d/dt h(t) = 196 - 9.8t = 0, this happens at t = 20. We then test for Maxima or Minima by taking the second derivative value of h(t) at t = 20. If it's positive, the value represents a Minimum; and if it's negative, the value represents a Maximum.

d/dt[d/dt h(t)] = -9.8, negative, irrespective of the value of t, hence the height is maximum at t = 20, and the real maximum height

h(20) = 1600 + 196(20) - 4.9(20)^2

h(20) = 1600 + 3920 - 1960 = 3560 meters.

[spoiler]Why our answers different?[/spoiler]
You're absolutely right, I incorrectly factored -4.9t² + 196t at the very beginning of my solution. My bad.
I've edited my response.

Cheers,
Brent
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by navinCH » Wed Jan 29, 2014 11:33 pm
Brent@GMATPrepNow wrote:
sanju09 wrote: The discussion that follow has nothing to do with the GMAT, so please ignore.

Hi Brent,

Excellent! Thanks for reminding completion of square technique. It is always possible in a quadratic expression. I tried it using Differential Calculus approach:

Differentiate h(t) = 1600 + 196t - 4.9t^2 w.r.t. t:

d/dt h(t) = 196 - 9.8t.

For Maxima and Minima d/dt h(t) = 196 - 9.8t = 0, this happens at t = 20. We then test for Maxima or Minima by taking the second derivative value of h(t) at t = 20. If it's positive, the value represents a Minimum; and if it's negative, the value represents a Maximum.

d/dt[d/dt h(t)] = -9.8, negative, irrespective of the value of t, hence the height is maximum at t = 20, and the real maximum height

h(20) = 1600 + 196(20) - 4.9(20)^2

h(20) = 1600 + 3920 - 1960 = 3560 meters.

[spoiler]Why our answers different?[/spoiler]
You're absolutely right, I incorrectly factored -4.9t² + 196t at the very beginning of my solution. My bad.
I've edited my response.

Cheers,
Brent
Thank you Brent Sir for acknowledging my point. I am an old student of Sanjeev Sir and I am extremely surprised why he didn't remember the completion of square technique that he only taught me one day, and made such an irrelevant DS problem.I am trying to contact him but he's not available on the numbers displayed here. I wish he'd respond here some day and and make his stand.
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