Vince’s family runs three lemonade stands during local events. Each stand serves lemonade in small, medium, and large glasses. T
he table gives the number of glasses of each size sold in each stand. Small Medium Large Stand 1 small 12 medium 11 large 6 Stand 2 small 10 medium 6 large 1 Stand 3 small 0 medium 3 large 9 The total collected from each stand was: stand 1, $87; stand 2, $43; and stand 3, $54. What is the price of a medium glass of lemonade? A. $2 B. $3 C. $4.50 D. $5
2 answers:
Answer:
B. $3
Step-by-step explanation:
Let s, m, l represent the prices of small, medium, and large lemonades, respectively. Three equations can be written based on the given revenue:
- 12s +11m +6l = 87
- 10s +6m +1l = 43
- 0s +3m +9l = 54
We can use the last equation to write an expression for l:
9l = 54 -3m . . . subtract 3m
l = 6 -m/3 . . . . . divide by 9
Substituting this into the first two equations gives ...
12s +11m +6(6 -m/3) = 87 . . . . . sub for l in first equation
12s +9m = 51 . . . . subtract 36, collect terms
Dividing by 3 gives
4s +3m = 17
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10s +6m +(6 -m/3) = 43
10s + (5 1/3)m = 36 . . . . subtract 6, collect terms
30s +16m = 108 . . . . . . multiply by 3
Dividing by 2 gives
15s + 8m = 54
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We can multiply the first of these by 15 and the second by -4, then add the results to eliminate the s variable.
15(4s +3m) -4(15s +8m) = 15(17) -4(54)
60s +45m -60s -32m = 255 -216 . . . . . eliminate parentheses
13m = 39 . . . . . collect terms
m = 3 . . . . . . . . divide by 13
The price of a medium glass of lemonade is $3.
_____
Most graphing calculators and available on-line calculators can make solving three equations in three unknowns trivially simple.
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This must be true for some value of x, since we have a quantity squared yielding a positive number, and since the equation is of second degree,there must exist 2 real roots.
Well he started off correct to the point of completing the square.