Question

Stuck on this. 60 points!

Answer 1

Answer:

$h(t)= (t-5)(t-11)$

The swimmer comes back up 11 seconds after the timer is started.

The swimmer dives into the water 5 seconds after the timer is started.

Step-by-step explanation:

Given function:

$h(t)=t^2-16t+55$

To factor a quadratic in the form $ax^2+bx+c$,

find two numbers that multiply to $ac$ and sum to $b$:

$\implies ac=55$

$\implies b=-16$

Two numbers that multiply to 55 and sum to -16 are: -11 and -5

Rewrite b as the sum of these two numbers:

$\implies t^2-11t-5t+55$

Factorize the first two terms and the last two terms separately:

$\implies t(t-11)-5(t-11)$

Factor out the common term (t - 11):

$\implies (t-5)(t-11)$

Therefore, the given formula in factored form is:

$h(t)= (t-5)(t-11)$

The swimmer's depth is modeled as h(t).  Therefore, when h(t) = 0 the swimmer will be at the surface of the water.

$\implies h(t)=0$

$\implies (t-5)(t-11)=0$

$\implies t-5=0\implies t=5$

$\implies t-11=0 \implies t=11$

Therefore, the swimmer will be at the surface of the water at 5 s and 11 s.

The swimmer's maximum depth is the vertex of the function. The x-value of the vertex is the midpoint of the zeros.  Therefore, the x-value of the vertex is t = 8.

Substitute t = 8 into the function to find the maximum depth:

$\implies h(8)=8^2-16(8)+55=-9$

So the swimmer's maximum depth is 9 ft.

True Statements

The swimmer comes back up 11 seconds after the timer is started.

The swimmer dives into the water 5 seconds after the timer is started.

Answer 2

Answer:

Statement 1 and 2

Step-by-step explanation:

If you take t^2 - 16t + 55 and find some of its graphical values, you will get:

Turning point: (8,-9)

Roots: (5,0) and (11,0)

When this graph is plotted and you imagine the x axis to be time (as stated in the question), each of the roots (x - intercept) must be when the swimmer goes under and when they come back up.

This means that the swimmer dived under the water at 5 seconds and came back up at 9, making the first 2 statements correct.

Now the fourth statement is ruled out.

The fifth statement is not plausible as the graph would have to have more than 2 roots for the swimmer to enter the water twice.

That leaves the third statement. If you imagine the depth of the swimmer to be the y axis of our imaginary graph, and we know that the y axis of the turning point is -9, that means that the swimmer's deepest dive was 9 feet under the water, ruling out the third statement too.

Hope this helps :D