I just have one more question.

I need to find the equation for the following word problem: a group of yeasts cells grows by 75% every 3 hours. At 9 a.m., there are 200 yeast cells.
Write an equation that models the number of yeast cells after 9 a.m.

So far I've worked out that C=200(1.75)^ , but my book says that it should be ^t/3, I suppose the /3 refers to the 'every three hours' bit, but, I don't understand why. Can someone please help me understand this?

the rate is 1.74 each three hours. THe way you write it then, giving t in hours, is just t/3. At t=3hrs, it give you

200*(1.75)^(3/3) or 200*1.75

yes, but why is it t/3?

Sure! Let's break down the problem and explain how to find the correct equation.

First, you correctly identified that the growth rate is 75% every 3 hours. This means that every 3 hours, the number of yeast cells is multiplied by 1.75.

To determine the equation that models the number of yeast cells after 9 a.m., we need to take into account the time. The growth rate is based on a 3-hour interval, so we need to express time in terms of this interval.

Let's denote the number of 3-hour intervals that have passed as t. If we divide the total time since 9 a.m. by 3, we can find the value of t. For example, if it's 9 a.m., then t would be 0. If it's 12 p.m., then t would be 1, and so on.

Now, let's write the equation using your initial formulation: C = 200(1.75)^t.

However, this equation assumes that the time passed is counted in the same units as the growth rate, which is "every 3 hours." To account for this, we need to include the factor of t/3 as an exponent for 1.75.

Therefore, the correct equation becomes: C = 200(1.75)^(t/3).

By incorporating the t/3 term, we ensure that the growth rate is applied appropriately for every 3-hour interval, as stated in the word problem.

I hope this explanation clarifies why the exponent should be t/3 in the equation! Let me know if you need any further assistance.