Science & Mathematics » Mathematics » A 16-oz jar of peanut butter in the shape of a right circular cylinder is 5in high and 3in in diameter and sells for \$1.60.?

# A 16-oz jar of peanut butter in the shape of a right circular cylinder is 5in high and 3in in diameter and sells for \$1.60.?

In the same store, a 26-oz jar of the same brand is 5&1/2in high and 3&1/4in in diameter. If the cost is directly proportional to volume, what should the price of the larger jar be? If the cost is directly proportional to weight, what should the price of the larger jar be?

• The small jar is 3*5 = 15 cu in. The larger jar is 3.25*5.5 = 17.9 cu in.
The small jar is 26/15= 1.7 oz/ cu in and \$1.60/15 = .11 \$ / cu in.
The larger jar is 17.9 cu in. 1.7*17.9 =30 oz and 17.9*.11 = \$1.97
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• Cylinder V = πr²h

case 1: V1 = π1.5²5 = 35.3 in³
case 2: V1 = π(3.25/2)²5.5 = 45.6 in³
ratio is 45.6/35.3 = 1.29
so the price would be 1.29•1.60 = \$2.07

by weight:
(assuming 16 oz and 26 oz are weight ounces)
ratio is 26/16 = 1.625
1.625•1.60 = \$2.60

PS a note about formats: you want a space character between the number and it's units, not a dash.
eg: 5 1/2 in, 16 oz
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• The formula for the volume of a right circular cylinder is shown below.
V = π * r^2 * h
For the 16 oz jar, V = π * 1.5^2 * 5 = π * 11.25 in^3
This is approximately 35.3cubic inches. To determine the cost per cubic inch divide \$1.60 by this number.

Cost per cubic inch = \$1.60 ÷ (π * 11.25)
This is approximately \$0.045. Let’s determine the volume of the larger jar.
r = ½ * 3¼ = 1.625 inches
V = π * 1.625^2 * 5.5 = π * 14.5234375 in^3
This is approximately 45.6 cubic inches.

Cost = π * 14.5234375 * 1.60 ÷ (π * 11.25)
This is approximately \$2.07.

If the cost is directly proportional to weight, what should the price of the larger jar be?
This is much simpler. Let’s use a proportion.

26 : 16 = x : \$1.60
x = 26/16 * 1.6 = \$2.60
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• i) 1.6 * 3.25^2*5.5/45 = \$2.07
ii) 1.6*13/8 = \$2.60
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• A 16-oz jar of peanut butter
in the shape of a right circular cylinder is 5 in high and 3 in in diameter and sells for \$1.60.
In the same store, a 26-oz jar of the same brand is 5.5 in high and 3.25 in in diameter.
If the cost is directly proportional to volume, what should the price of the larger jar be?
Volume of a right circular cylinder: πr^2h
r = d/2 = 1.5 inches
h = 5 inches
Cost = (πr^2h)k, where k is the constant of proportion
1.60 = π(1.5)2(5)k
k = 1.6/[π(1.5)2(5)] = 0.045271
So the formula is: Cost = 0.045271(πr^2h)
For 5.5 inches high, and 3.25 inches diameter:
r = 1.625
Cost = π(1.625)2(5.5)(0.045271) = \$2.07
Weight: Cost = kw; where w is weight and k is constant of proportionality
1.6 = 16k
k = 1.6/16
k = 0.10
Cost = 0.10w
c = 0.10(26) = \$2.60
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