Dr. Silas studies a culture of bacteria under a microscope. The function b(t)=50(1.4)^t represents the number of bacteria t hours after Dr. Silas begins her study.a) What is the y-intercept of the function? What does this mean in the context of the problem?b) Is this exponential growth or decay? How do you know?c) How many bacteria would there be 5 hours after Dr. Silas began her study?Please show all your work!

a) Remember that the y-intercept of a exponential function [tex]f(x)=a^{x} [/tex] is [tex]f(0)[/tex], so the only thing to do to find the y-intercept in our functions is evaluate it at t=0:
[tex]b(t)=50(1.4) ^{t} [/tex]
[tex]b(0)=50(1.4)^{0} [/tex]
[tex]b(0)=50(1)[/tex]
[tex]b(0)=50[/tex]
We can conclude that the y-intercept of our function is (0,50), and it represents the initial bacteria population in the sample.

b) To find if the function is growing or decaying, we are going to convert its decimal part to a fraction. Then, we will compare the numerator and the denominator of the fraction. If the numerator is grater than the denominator, the function is growing; if the opposite is true, the function is decaying.
Remember that to convert a decimal into a fraction we are going to add the denominator 1 to our decimal and then we'll multiply both of them by a power of ten for each number after the decimal point:
[tex] \frac{1.4}{1} . \frac{10}{10} = \frac{14}{10} = \frac{7}{5} [/tex]
Now we can rewrite our exponential function:
[tex]b(t)=50( \frac{7}{5})^{t} [/tex]
Since the numerator is grater than the denominator, it is growing faster than the denominator; therefore the function is growing.

c) The only thing we need to do here is evaluate the function at t=5:
[tex]b(t)=50(1.4)^{t} [/tex]
[tex]b(5)=50(1.4)^{5} [/tex]
[tex]b(5)=50(5.37824)[/tex]
[tex]b(5)=268.912[/tex]

We can conclude that after 5 hours Dr. Silas began her study will be 268.9 bacteria in the sample.