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Please help with this calculus IVT problem! (I need help specifically with part c)

Please help with this calculus IVT problem! (I need help specifically with part c-example-1

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Part (a)

We'll use the second piece of the piecewise function.

The leading terms in the numerator and denominator are 500t and t respectively. Their ratio is (500t)/(t) = 500.

As t heads to infinity, the fraction (500t-1500)/(t-1) approaches 500.

Therefore, the limiting value is 500.

It means the amount of money raised approaches $500 as time continues to march on forever. We'll never actually reach 500 itself. Rather we just get closer and closer.

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Part (b)

Plug t = 4 into the first piece

f(t) = 5*2^(t+1)-3

f(4) = 5*2^(4+1)-3

f(4) = 157

Do the same for the second piece

g(t) = (500t-1500)/(t-1)

g(4) = (500*4-1500)/(4-1)

g(4) = 166.6667 approximately

Because f(4) = g(4) is not a true statement, this would mean F(t) is not continuous at t = 4. We have a jump discontinuity.

Use graphing software such as Desmos or GeoGebra to confirm this.

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Part (c)

Replace M(t) with 100 and solve for t.

M(t) = 210(2^t-1)/(2^t+5)

100 = 210(2^t-1)/(2^t+5)

100(2^t+5) = 210(2^t-1)

100(w+5) = 210(w-1) ............... let w = 2^t

100w+500 = 210w-210

100w-210w = -210-500

-110w = -710

w = -710/(-110)

w = 6.45454545 approximately

2^t = 6.45454545

log(2^t) = log(6.45454545)

t*log(2) = log(6.45454545)

t = log(6.45454545)/log(2)

t = 2.6903154998514 approximately

It takes about 2.6903 days to reach $100

This value of t is in the interval 0 ≤ t ≤ 4, which makes it a valid solution.

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