How do I answer this calculus question?

Question

The quantity 𝑄 of a substance in grams that is decaying exponentially is given by the formula

𝑄(𝑡) = 35(. 88)^t

where 𝑡 is the time in hours.

How much is left after 5 hours?

How fast is the substance decaying after 5 hours?

Use these values to estimate the quantity remaining after 7 hours.

Answer 1

How much is left after 5 hours

Q(5) will give you the amount of substance left after 5 hours.

Q(5) = 35(.88)⁵ = 18.5

How fast is the substance decaying after 5 hours?

Recall that the rate of change of a function is given by its derivative, so Q'(t) will describe the how quickly the amount of substance is changing at a time t. Q'(5) will give you the answer.

Q'(t) = 35*d(.88^t)/dt = 35*(ln(.88)*.88^t)

Q'(5) = 35*(ln(.88)*.88⁵) = -2.4

Use these values to estimate the quantity remaining after 7 hours.

If we know the value of Q and Q' at a point t0, we can estimate the value of Q at neighboring t values with the formula Q(t) ≈ Q(t0) + Q'(t0)*(t-t0). In our case, we know t0=5, and we want to know Q(7).

Q(7) ≈ Q(5) + Q'(5)*(7-5)

Q(7) ≈ 18.5 + (-2.4)*(2)

Q(7) ≈ 13.7

Hope this helps!

Answer 2

It looks like t=5...............................