Question

The graph shows the amount of a medicine m, in milligrams, remaining in a patient’s body h hours after receiving an injection. The amount of the medicine decreases exponentially.

By what factor did the medicine decrease in the first hour and a half?
By what factor did the medicine decrease in the first half hour?
What about in the first hour?

Answer 1

a) The medicine decrease in the hour and half at a factor of 0.44

b) The medicine decrease in the first half hour, and the first hour at a factor of 0.44

c) The equation relating m to h is y=270*0.44^h

An exponential function is a function defined by
`y = ab^x`, where a represents the initial value, and b represents the rate

(a) The factor at which the medicine decreased, in 1.5 hour

From the graph, we have the following ordered pairs

(x,y) = (0,270) and (1.5,80)

Given that:
`y=ab^x`

At point (0,270), we have:

`ab^0=270`

a=270

At point (1.5,80), we have:

`ab^(1.5)=80`

Substitute 270 for a:

`270b^(1.5)=80`

Divide both sides by 270:

`b^(1.5)=0.2963`

Take 1.5th root of both sides:

b=0.44

Hence, the medicine decrease in the hour and half at a factor of 0.44

(b) The factor at which the medicine decreased in the first half hour, and the first hour

In (a), we have:

The decay factor (b) to be 0.44

This represents the factor at which the medicine decreased throughout.

Hence, the medicine decrease in the first half hour, and the first hour at a factor of 0.44

(c) The exponential equation

In (a), we have: a=270, b=0.44

So, the exponential equation is:

In terms of m and h, we have:

`y=270*0.44^x`

Hence, the equation relating m to h is
`y=270*0.44^h`