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A batch of widgets costs p + 15 dollars for a company to pro

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suk1234
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A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   Updated on: 08 Jan 2014, 05:20
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A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 – p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
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B

Originally posted by suk1234 on 08 Jan 2014, 04:14.
Last edited by Bunuel on 08 Jan 2014, 05:20, edited 1 time in total.
Renamed the topic and edited the question.
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   08 Jan 2014, 05:30
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suk1234 wrote:
A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 – p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


Profit = Revenue - Cost = \(p(9 - p) - (p + 15) = -p^2 + 8p - 15 = -(p-3)(p-5)\).

At this point we can substitute the options to see for which value is \(-(p-3)(p-5)>0\) OR solve \(-(p-3)(p-5)>0\).

Answer: B.

If interested in solving \(-(p-3)(p-5)>0\):

\(-(p-3)(p-5)>0\) --> \((p-3)(p-5)<0\) --> the "roots" are 3 and 5. "<" sign indicates that the solution lies between the roots: \(3<p<5\). The only option within this range is B.

For more about this technique check here: if-x-is-an-integer-what-is-the-value-of-x-1-x-2-4x-94661.html#p731476

Hope this helps.
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   02 Mar 2017, 23:52
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suk1234 wrote:
A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 – p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


The options give the values of p. Just plug them in to find the value for which p(9 - p) (= revenue) is more than p + 15 (= cost).
When you put p = 3, 3*(9 - 3) = 18 and 3 + 15 = 18. So revenue is not higher.
When you put p = 4, 4*(9 - 4) = 20 and 4 + 15 = 19. Revenue is higher.

Answer (B)
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   08 Jan 2014, 05:56
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suk1234 wrote:
A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 – p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


to make profit, selling price should be greater than cost price
p(9-p) - p-15 >0
=> p^2-8*p+15<0
(p-3)*(p-5)<0
hence p=4
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   02 Mar 2017, 12:13
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Profit equals revenue minus cost. The company's profit is:
p(9 – p) – (p + 15) = 9p – p2 – p – 15
= – p^2 + 8p – 15
= –(p^2 – 8p + 15)
= –(p – 5)(p – 3)
Profit will be zero if p = 5 or p = 3, which eliminates answers (A) and (C). For p > 5, both (p – 5)
and (p – 3) are positive. In that case, the profit is negative, i.e. the company loses money. The profit
is only positive if (p – 5) and (p – 3) have opposite signs, which occurs when 3 < p < 5.
The correct answer is B.
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   07 Mar 2017, 07:37
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suk1234 wrote:
A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 – p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


Since profit = revenue - cost, or P = R - C, we need to determine a value of p such that revenue > cost. We have that p + 15 = cost and p(9 - p) = revenue. Therefore, the profit is P = p(9 - p) - (p + 15) = 9p - p^2 - p - 15 = -p^2 + 8p - 15.

Notice that the graph of P = -p^2 + 8p - 15 is a parabola that opens downward, so the maximum value is at its vertex. Recall that we can use the formula -b/2a to find the x-value (in this case, the p-value) of the vertex:

p = -b/2a = -8/(2(-1)) = -8/-2 = 4

Answer: B
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   13 Mar 2015, 02:44
Just to be clear after

(p-3)*(p-5)<0

we have two options:
p<3 and p>5, no solution because no interceptions of intervals
p>3 and p<5, so 3<p<5

only B
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   17 Oct 2016, 10:43
I have a few questions about this...

When it was time to factor the quadratic I got -p²+8p-15>0. I then divided by -1 so I got p²-8p+15 then got (p-3) and (p-5) so 3 and 5 cannot be the answers. Then you get p²-8p+15<0 so don't we get p<3, p<5

If we try to solve it isn't it p(p-9)-(p+15)? When I plug in 4 the answer is 0.


@temurkohn


(p-3)*(p-5)<0

we have two options:
p<3 and p>5, no solution because no interceptions of intervals
p>3 and p<5, so 3<p<5

How did you get to p>3?

You have p-3<0, wouldn't it be p<3?
Help?
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   17 Oct 2016, 11:00
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joannaecohen wrote:
I have a few questions about this...

When it was time to factor the quadratic I got -p²+8p-15>0. I then divided by -1 so I got p²-8p+15 then got (p-3) and (p-5) so 3 and 5 cannot be the answers. Then you get p²-8p+15<0 so don't we get p<3, p<5

If we try to solve it isn't it p(p-9)-(p+15)? When I plug in 4 the answer is 0.


@temurkohn


(p-3)*(p-5)<0

we have two options:
p<3 and p>5, no solution because no interceptions of intervals
p>3 and p<5, so 3<p<5

How did you get to p>3?

You have p-3<0, wouldn't it be p<3?
Help?


There are few things that you need to know.

(x-a)(x-b) < 0 means a<x<b (provided a<b)

(x-a)(x-b) > 0 means x<a or x>b (provided a<b)

Now, try solving the question. You will get the answer.

Also, even if you substitute p = 4, you will get the value of profit = p(9 – p) - (p + 15 ) = 4(9-4) - (4+15) = 20-19=1.
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   09 Sep 2019, 07:53
suk1234 wrote:
A batch of widgets costs p + 15 dollars for a company to produce and each batch sells for p(9 – p) dollars. For which of the following values of p does the company make a profit?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


For Profit CP < SP

So, \(p + 15 < p(9 – p)\)

Or, \(p + 15 < 9p – p^2\)

Now , plug in the values -

(A) 18 < 27 - 9 => 18 < 18 (Not Possible)
(B) 19 < 36 - 16 (Possible)
(C) 20 < 45 - 25 (Not Possible)
(D) 21 < 54 - 36 (Not Possible)
(E) 22 < 63 - 49 (Not Possible)

Hence Answer must be (B)
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Re: A batch of widgets costs p + 15 dollars for a company to pro [#permalink] New post   13 May 2023, 01:34
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