The current circulation of a particular magazine is 3,000 copies per week. The editor projects a growth rate of
g(t) = 4 + 5t^2/3
copies per week after t weeks.
a. Find the circulation function based on this projection.
b. Find the circulation in 2 years.
a. To find the circulation function based on the given growth rate, we need to integrate the growth rate function g(t) with respect to t.
So, let's integrate g(t):
∫(4 + 5t^(2/3)) dt
To integrate t^(2/3) with respect to t, we can use the power rule of integration, which states that for any real number n ≠ -1:
∫t^n dt = (t^(n+1))/(n+1)
Applying the power rule:
= ∫4 dt + ∫5t^(2/3) dt
= 4t + (5(t^(2/3 + 1)) / (2/3 + 1))
= 4t + (5(t^(5/3)) / (5/3))
= 4t + (15t^(5/3)) / 5
= 4t + 3t^(5/3)
Therefore, the circulation function based on the given projection is:
C(t) = 4t + 3t^(5/3)
b. To find the circulation in 2 years (t = 2), we substitute t = 2 into the circulation function C(t):
C(2) = 4(2) + 3(2)^(5/3)
= 8 + 3(2)^(5/3)
Now, let's simplify the expression:
= 8 + 3(2^(5/3))
= 8 + 3(2^(5/3))
≈ 8 + 14.12
≈ 22.12
Therefore, the circulation in 2 years is approximately 22.12 copies per week.
a. To find the circulation function based on the given projection, we need to integrate the growth rate function, g(t).
The circulation function, C(t), can be represented as the integral of the growth rate function:
C(t) = ∫(4 + 5t^(2/3)) dt
To integrate the function, we can use the power rule for integration:
∫(t^n) dt = (t^(n+1))/(n+1)
Applying this rule to the given function, we have:
C(t) = ∫(4 + 5t^(2/3)) dt
= 4t + (5/(2/3 + 1)) * t^(2/3 + 1)
= 4t + (5/(5/3)) * t^(5/3)
= 4t + (15/5) * t^(5/3)
= 4t + 3t^(5/3)
= 4t + 3∛(t^5)
So, the circulation function is C(t) = 4t + 3∛(t^5).
b. To find the circulation in 2 years, we substitute t = 2 into the circulation function:
C(2) = 4(2) + 3∛((2^5))
= 8 + 3∛(32)
= 8 + 3(2)
= 8 + 6
= 14
Therefore, the circulation in 2 years is 14,000 copies per week.
Let C(x) be the circulation function
given: C ' (x) = 4+5t^(2/3)
then
C(x) = 4t + 3t^(5/3) + c, where c is a constant
when x = 0 (now) , C(0) = 3000
3000 = 0 + 0 +c
so C(x) = 4t + 3t^(5/3) + 3000
so in 2 years, x = 2
C(2) = 8 + 3(2)^(5/3) + 3000
= appr 3018