Question

A sports club was formed in the month of May last year. The function below, M(t), models the number of club members for the first 10 months, where t represents the number of months since the club was formed in May. m(t)=t^2-6t+28 What was the minimum number of members during the first 10 months the club was open? A. 19 B. 28 C. 25 D. 30

Answer 1

A: 19

Explanation:

For this, we can complete the square. We first look at the first 2 terms,

t^2 and -6t.

We know that
`(t-3)^2` will include terms.

`(t-3)^2 = t^2 - 6t + 9`

But
`(t-3)^2` will also add 9, so we can subtract 9. Putting this into the equation, we get:

`m(t) = (t-3)^2 - 9 +28`

`m(t) = (t-3)^2 +19`

Using the trivial inequality, which states that a square of a real number must be positive, we can say that in order to have the minimum number of members, we need to make (t-3) = 0. Luckily, 3 months is in our domain, which means that the minimum amount of members is 19.