Salmonella bacteria, found on almost all chicken and eggs,grow rapidly in a nice warm place. If just a few hundredbacteria are left on the cutting board when a chicken is cut up,and they get into a potato salad, the population beginscompounding. suppose the number present in the potato saladafter x is given by

f(x) = 500 x 2^3x
a) if the potato salad is left out on the table, how manybacteria are present 1 hour later?
b. how many were present initially?
c. how ofter do the bacteria double?
d. how quickly will the number of bacteria increase to32,000?

no ideas on any of the questions? They gave you the function, so things should just fall right out.

To answer these questions, we can use the given function f(x) = 500 * 2^(3x), where x represents time in hours. Let's go through each question one by one:

a) If the potato salad is left out on the table for 1 hour, we need to find f(1). Plugging in x = 1 into the function, we get:
f(1) = 500 * 2^(3*1)
= 500 * 2^3
= 500 * 8
= 4000

So, 4000 bacteria would be present in the potato salad 1 hour later.

b) To find the initial number of bacteria present, we need to find f(0) as we are considering the initial time. Plugging in x = 0 into the function, we get:
f(0) = 500 * 2^(3*0)
= 500 * 2^0
= 500 * 1
= 500

Therefore, initially, there would be 500 bacteria present in the potato salad.

c) To find how often the bacteria double, we need to find the value of x when the function doubles. In other words, we need to solve the equation:
f(x) = 2 * f(x-1)

Substituting the function f(x) = 500 * 2^(3x), we get:
500 * 2^(3x) = 2 * (500 * 2^(3(x-1)))

Simplifying the equation:
2^(3x) = 2^(3(x-1))

By equating the exponents:
3x = 3(x-1)

Solving for x:
3x = 3x - 3
0 = -3

This equation has no solution, meaning the bacteria do not double in population based on the given function. Therefore, the bacteria do not double in this scenario.

d) To find how quickly the number of bacteria will increase to 32,000, we need to find the value of x when f(x) = 32,000. Setting up the equation:
500 * 2^(3x) = 32,000

Dividing both sides by 500:
2^(3x) = 64

Taking the logarithm (base 2) of both sides to solve for x:
3x = log2(64)
3x = 6

Dividing both sides by 3:
x = 2

So, the number of bacteria will increase to 32,000 after 2 hours.

In summary:
a) 4000 bacteria are present 1 hour later.
b) Initially, there are 500 bacteria present.
c) The bacteria do not double in this scenario.
d) The number of bacteria will increase to 32,000 after 2 hours.