ted sells his homemade peanut butter for 1.60 a jar at the local farmers Market the jar is 8 cm in diameter and 10 cm high he decides he will also sell peanut butter in jats that are 16 cm in diameter and 20 cm high.What should he charge if he uses the same price per cubic centimeter
volume of current jar
= π(4^2)(10) = 160π cm^3
new jar:
volume = π(8^2)(20) = 1280π
set up a ratio, where x is the new cost
x/1.6 = 1280π/(160π)
x = 1.6(1280)/160 = 12.80
(notice the new jar is 8 times the volume of the smaller jar, so the cost is 8 times the original cost)
Ted sells his homemade peanut butter for
$1.60 a jar at the local Farmers’ Market.
The jar is 8 cm in diameter and 10 cm high.
He decides he will also sell peanut butter in
jars that are 16 cm in diameter and 20 cm
high. What should he charge if he uses the
same price per cubic centimetre?
To determine the price of the new jar, we need to compare the volumes of the two jars and adjust the price accordingly.
Let's calculate the volume of each jar:
Volume of the original jar (8 cm diameter and 10 cm high):
V1 = π * (D1/2)^2 * H1
= π * (8/2)^2 * 10
= 160π cm^3
Volume of the new jar (16 cm diameter and 20 cm high):
V2 = π * (D2/2)^2 * H2
= π * (16/2)^2 * 20
= 640π cm^3
Now, we can calculate the new price per cubic centimeter:
Price per cm^3 = 1.60 / V1
Therefore, the price for the new jar can be determined by multiplying the volume of the new jar by the price per cubic centimeter:
Price for new jar = Price per cm^3 * V2
= (1.60 / V1) * V2
Now, let's substitute the values into the equation:
Price for new jar = (1.60 / 160π) * 640π
= (1.60 / 160) * 640
= 0.016 * 640
= $10.24
Therefore, Ted should charge $10.24 for the new jar if he wants to use the same price per cubic centimeter.
To answer this question, we need to calculate the volume of each jar and then compare the volumes to determine the new price for the larger jar.
1. Calculate the volume of the small jar:
The formula for the volume of a cylinder is V = πr^2h, where:
- V = Volume
- π ≈ 3.14159 (pi)
- r = radius of the jar (diameter/2)
- h = height of the jar
Given values for the small jar:
- Diameter = 8 cm
- Radius (r) = 8/2 = 4 cm
- Height (h) = 10 cm
Using the formula: V_small = π(4^2)(10)
Calculating: V_small = 3.14159 * 16 * 10 = 502.65408 cubic cm (approximately)
2. Calculate the volume of the large jar:
Using the same formula for the volume of a cylinder, but this time with the given dimensions for the large jar:
- Diameter = 16 cm
- Radius (r) = 16/2 = 8 cm
- Height (h) = 20 cm
Using the formula: V_large = π(8^2)(20)
Calculating: V_large = 3.14159 * 64 * 20 = 12056.74848 cubic cm (approximately)
3. Find the ratio of the volumes:
To find the ratio of the volumes, divide the volume of the large jar by the volume of the small jar:
Ratio = V_large / V_small
Calculating: Ratio = 12056.74848 / 502.65408 ≈ 23.98 (approximately)
4. Determine the new price:
Since we know that Ted charges the same price per cubic centimeter for both jars, we can multiply the ratio of the volumes by the price of the small jar to find the new price for the large jar:
New price = Ratio * Price_small
Given: Price_small = $1.60
Calculating: New price = 23.98 * $1.60 ≈ $38.37 (approximately)
Therefore, Ted should charge approximately $38.37 for the larger jar if he uses the same price per cubic centimeter.