Jake stores a small cache of4-inch-diameter snowballs in the basement freezer, unaware that the freezer’s self-defrosting feature will cause each snowball to lose about 1 cubic inch of volume every 40 days. Here members them a year later (call it 360 days) and goes to retrieve them.(𝑉=43𝜋𝑟3)
By what fraction is the volume reduced after 360 days?
A) ½
B) 4/3
C) 8/9
D) 36/4
E) 1/9
To find the fraction by which the volume is reduced after 360 days, we need to compare the initial volume to the volume after 360 days.
The initial volume of each snowball is given by the formula V = (4/3)πr^3, where r is the radius of the snowball. In this case, the diameter is 4 inches, so the radius, r, is 4/2 = 2 inches.
Substituting the value of r into the formula, we can calculate the initial volume:
V_initial = (4/3)π(2^3) = (4/3)π(8) = 32π
After 360 days, each snowball loses 1 cubic inch of volume every 40 days. Therefore, the snowballs have been in the freezer for 360/40 = 9 cycles of 40 days.
During each cycle of 40 days, one snowball loses 1 cubic inch of volume. So, after 9 cycles, it loses 9 cubic inches of volume.
The final volume after 360 days is given by V_final = V_initial - 9.
To find the fraction by which the volume is reduced, we need to calculate the ratio of the final volume to the initial volume:
Fraction = V_final / V_initial
Fraction = (V_initial - 9) / V_initial
Fraction = (32π - 9) / (32π)
To simplify the fraction, we can divide both the numerator and the denominator by 1/π:
Fraction = (32π - 9)/(32π) * (1/π)/(1/π)
Fraction = (32 - 9/π)/(32)
As we divide both numerator and denominator by π, we can approximate the value of π as 3.14:
Fraction ≈ (32 - 9/3.14)/(32)
Fraction ≈ (32 - 2.87)/(32)
Fraction ≈ 29.13/32
To get the fraction value, we divide both numerator and denominator by 29.13:
Fraction ≈ (29.13/29.13)/(32/29.13)
Fraction ≈ 1/(32/29.13)
Fraction ≈ 1/(1.10)
Fraction ≈ 0.909
Therefore, the fraction by which the volume is reduced after 360 days is approximately 0.909. This is equivalent to 9/10 or 8/9.
So, the correct answer is (C) 8/9.