How many ounces of each juice should a glass of the blend contain if it is to me
ID: 524642 • Letter: H
Question
How many ounces of each juice should a glass of the blend contain if it is to meet the minimum vitamin requirement while containing a minimum number of calories? The owner of the Health JuiceBar wishes to prepare a low-calorie fruit juice with a high vitamin-A and -C content by blending orange juice and pink grapefruit juice. Each glass of the blended juice is to contain at least 1190 International Units (U) of vitamin A and 152 IU of vitamin C.One ounce of orange juice contains 70 TU of vitamin A, 16 IU of vitamin C, and 14 calories; each ounce of pink grapefruit juice contains 140 IU of vitamin A, 12 TU of vitamin C, and 12 calories. How many ounces of each juice should a glass of the blend contain if it is to meet the minimum vitamin requirements while containing a minimum number of calories?Explanation / Answer
We will have the minimum calories if we have more of the grapejuice (contains less calories). Let we mix x ounce of grapefruit juice with y ounce of orange juice.
Therefore, x ounce of grapefruit juice will have (x*140) IU Vit. A and (x*12) IU Vit.C.
y ounce of orange juice will contain (y*70) IU Vit. A and (y*16) IU Vit. C.
Thus, we must have
140x + 70y = 1190
===> 70*(2x + y) = 1190
===> 2x + y = 17 …..(1)
12x + 16y = 152
===> 4*(3x + 4y) = 152
===> 3x + 4y = 38 ….(2)
From (1), y = 17 – 2x
Plug in this expression for y in (2) and obtain
3x + 4*(17 – 2x) = 38
===> 3x + 68 – 8x = 38
===> 68 – 5x – 38 = 0
===> 30 – 5x = 0
===> x = 30/5 = 6
Put x = 6 in (1) and obtain
2*6 + y = 17
===> 12 + y = 17
===> y = 5
The owner must mix 6 ounces of orange juice with 5 ounces of grapejuice (ans).
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