two spherical cantaloupes of the same kind are sold at a fruit and vegetable stand. The circumference of one is 60cm and that of the other is 50cm. The larger melon is 1 1/2 times as expensive as the smaller. Which melon is the better buy and why?
C = pi * d
60 = 3.14d
60/3.14 = d
19.1 = d
V = 4/3 * pi * r^3
V = 4/3 * 3.14 * 9.55^3
V = 4/3 * 3.14 * 871
V = 3646.6 sq. cm
Do the same for the smaller melon.
C=pi*d
50=3.14d
50/3.14=d
15.9=d
Still trying to get the V problems
To determine which melon is the better buy, we need to compare their prices relative to their sizes.
Step 1: Calculate the radius of each melon
The circumference of a sphere is related to its radius by the formula: circumference = 2π * radius.
So, dividing the circumference by 2π will give us the radius of each melon.
For the first melon:
Circumference = 60 cm
Radius = Circumference / (2π) = 60 cm / (2π) ≈ 9.55 cm
For the second melon:
Circumference = 50 cm
Radius = Circumference / (2π) = 50 cm / (2π) ≈ 7.96 cm
Step 2: Calculate the volumes of each melon
The volume of a sphere is given by the formula: volume = (4/3)π * (radius)^3.
For the first melon:
Volume = (4/3)π * (9.55 cm)^3 ≈ 3463.20 cm³
For the second melon:
Volume = (4/3)π * (7.96 cm)^3 ≈ 1708.43 cm³
Step 3: Compare the prices per unit volume
The larger melon is 1 1/2 times as expensive as the smaller melon.
Let's denote the price of the smaller melon as "P".
The price of the larger melon would then be (3/2)P.
To find which melon is the better buy, we need to compare the prices per unit volume (price/volume) for each melon.
For the first melon:
Price per unit volume = P / 3463.20 cm³
For the second melon:
Price per unit volume = (3/2)P / 1708.43 cm³
Step 4: Compare the price per unit volume of the melons
Since both melons have the same kind, we assume their quality is the same. Therefore, a lower price per unit volume indicates a better buy.
Compare the two price per unit volume values to determine which is smaller. The melon with the smaller price per unit volume is the better buy.
If (P / 3463.20 cm³) < ((3/2)P / 1708.43 cm³), then the smaller melon is the better buy.
Simplifying the inequality, we get:
(1 / 3463.20 cm³) < ((3/2) / 1708.43 cm³)
Calculating the left side gives approximately 0.000288, and the right side gives approximately 0.001042. Since 0.000288 < 0.001042, we can conclude that the smaller melon is the better buy.
Therefore, based on the comparison of their prices relative to their sizes, the smaller melon is the better buy.