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14 Sep 2022, 07:18
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Difficulty:
95%
(hard)
Question Stats:
54% (02:55) correct
46%
(02:44)
wrong
based on 186
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A shoe store sells a new model of running shoe for $32. At that price, the store sells 80 pairs each week. The manager however estimated that the store will sell 20 additional pairs of shoes each week for every $2 reduction in the price. According to these estimations, what is the range of the shoe prices, which would allow the store to maintain or increase its current weekly revenue from the shoes ?
A. 12
B. 13
C. 23
D. 24
E. 25
Check twin question HERE.
A. 12
B. 13
C. 23
D. 24
E. 25
Check twin question HERE.
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02 Apr 2024, 01:26
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Official Solution:
A shoe store sells a new model of running shoe for $32. At this price, the store sells 80 pairs each week. However, the manager estimated that for every $2 reduction in the price, the store will sell an additional 20 pairs of shoes per week. According to these estimations, what is the range of shoe prices that would allow the store to maintain or increase its current weekly revenue from the shoes?
A. 12
B. 13
C. 23
D. 24
E. 25
The current weekly revenue from the shoes is \(32 * 80\) dollars.
Assuming \(x\) is the number of times the price was reduced by 2 dollars, the price of a pair of shoes would be \(32 - 2x\) dollars, and a total of \(80 + 20x\) pairs of shoes would be sold, generating revenue equal to \((32 - 2x)(80 + 20x)\) dollars.
The question asks to find the range of shoe prices such that \((32 - 2x)(80 + 20x) \geq 32 * 80\). Reduce by \(40\):
\((16 - x)(4 + x) \geq 32*2\)
\(-x^2 + 12 x + 64 \geq 32*2\)
\(x^2 - 12x \leq 0\)
\(x(x - 12) \leq 0\)
The above inequality holds for \(0 \leq x \leq 12\). Thus, the shoe prices for which the store will maintain or increase its current weekly revenue start from \(32 - 2 * 0 = 32\) dollars and end with \(32 - 2 *12 = 8\) dollars. Therefore, the range of prices is \(32 - 8 = 24\).
Answer: D
A shoe store sells a new model of running shoe for $32. At this price, the store sells 80 pairs each week. However, the manager estimated that for every $2 reduction in the price, the store will sell an additional 20 pairs of shoes per week. According to these estimations, what is the range of shoe prices that would allow the store to maintain or increase its current weekly revenue from the shoes?
A. 12
B. 13
C. 23
D. 24
E. 25
The current weekly revenue from the shoes is \(32 * 80\) dollars.
Assuming \(x\) is the number of times the price was reduced by 2 dollars, the price of a pair of shoes would be \(32 - 2x\) dollars, and a total of \(80 + 20x\) pairs of shoes would be sold, generating revenue equal to \((32 - 2x)(80 + 20x)\) dollars.
The question asks to find the range of shoe prices such that \((32 - 2x)(80 + 20x) \geq 32 * 80\). Reduce by \(40\):
\((16 - x)(4 + x) \geq 32*2\)
\(-x^2 + 12 x + 64 \geq 32*2\)
\(x^2 - 12x \leq 0\)
\(x(x - 12) \leq 0\)
Answer: D
General Discussion
14 Sep 2022, 07:37
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Bunuel
Present weekly revenue = 80 x $32 = $2560
So according to estimation, Sales will increase by 20 pairs for every $2 decrease in price
So the possibilities are
100 x $30 = $3000
120 x $28 = $3360
140 x $26 = ....
.
..
.
.
.
200 x $20 = $4000 (Hereafter it starts decreasing)
.
.
.
.
..
.
320 x $8 = $2560 (Able to maintain the revenue)
So the range of shoe prices = $32 - $8 = $24
Ans is D
