please help me with calculus!?

Question

1.if the size of a baceria colony doubles in 6 hours how long will it take for the number of bacteria to triple?

2. the fourth route of 64 + 9 fourth route of 4 = ?

3. if f(t)=square route of (t^2-36) find all values of t for which f(t) is a real number.

solve f(t)=8

if u can solve any of these please please explain it to me!

Answer 1

(1) If it doubles in 6 hours, assuming exponential growth, the population size increase in x hours is given by:

2^(x/6)

To find out how long it takes to triple, solve:

2^(x/6) = 3
log2(2^(x/6)) = log2(3)
(x/6) = log2(3)
x = log2(3) * 6
x = ln(3)/ln(2) * 6
x = 9.5098

About 9.5 hours.

(2) 64^(1/4) + 9*(4^(1/4))

64^(1/4) =
(16*4)^(1/4) =
(16^(1/4)) * (4^(1/4)) =
((2^4)^(1/4)) * (4^(1/4)) =
2 * 4^(1/4)

Substituting that back in:

64^(1/4) + 9*(4^(1/4)) =
2 * 4^(1/4) + 9 * 4^(1/4) =
11 * 4^(1/4)

11 times the fourth root of 4.

(3) f(t) = sqrt(t^2 - 36)

To solve for f(t) real, the quantity inside the square root must be greater than or equal to zero:

t^2 - 36 >= 0
t^2 >= 36

The solution to that is t >= 6 OR t <= -6.

Basically everything EXCEPT t between -6 and 6.

Another way to work out the problem:

Factor (t^2 - 36) into (t + 6)(t - 6)

When t is greater than 6, both terms are positive and the product is positive. When t is less than -6, both terms are negative and the product is positive. When t is exactly 6 or -6, one of the terms is zero and the product is zero.

However, when t is between -6 and 6, t + 6 is positive and t - 6 is negative, so the product is negative.

Solve:

f(t) = 8
and
f(t) = sqrt(t^2 - 36)

Substitute, square both sides, and solve:
8 = sqrt(t^2 - 36)
64 = t^2 - 36
t^2 = 100

To solutions:
t = 10, t = -10