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Difficulty:
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43% (02:28) correct
57%
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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
A. 12
B. 13
C. 23
D. 24
E. 25
02 Apr 2024, 01:25
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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 \(40\):
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 \(40\):
\((16 - x)(4 + x) \geq 32*2\)
\(-x^2 + 12 x + 64 \geq 32*2\)
\(x^2 - 12x \leq 0\)
\(x(x - 12) \leq 0\)
\(-x^2 + 12 x + 64 \geq 32*2\)
\(x^2 - 12x \leq 0\)
\(x(x - 12) \leq 0\)
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
19 Apr 2024, 06:24
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Bunuel
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
BunnelA 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
Why are we reducing the LHS of the equation by 40.
