Question

1. The function represents the amount of a medicine, in mg, in a person's body hours after taking the medicine. Here is a graph of . ( the picture that i sent in ) a. How many mg of the medicine did the person take at the start?B. Complete the table c. Write an equation that defines .d. After 7 hours, how many mg of medicine remain in the person's

Answer 1

a. At start, the person took 80 mg

b. The complete table includes

x y

0 80

1 40

2 20

3 10

4 5

c. The equation for the graph is f(x) = 40(0.5)ˣ

d. 0.625 mg of the drug will remain after 7 hours

How to find the equation

An exponential function is a mathematical function of the form f(x) = a(b)ˣ

The parameters are defined as follows

the starting = a

the base = b

the exponents = x

Analyzing the function

f(x) = a(b)ˣ using (0, 80)

80 = a - 1 (b exponent 0 is 1)

a = 80

Solving for B using point (2, 20)

20 = 80 * (b)²

b² = 20/80

b = √1/4

b = 0.5

For x = 1

y = 80 * (0.5) = 40

For x = 3

y = 80 * (0.5)³ = 10

For x = 7 (after 7 hours)

y = 80 * (0.5)⁷ = 0.625

Answer 2

• (a)80 mg

,

• (c)f(t)=80(0.5)^t

,

• (d)0.625 mg

Explanation:

Part A

From the point (0, 80), we see that the person took 80 mg of the medicine at the start.

Part B

From the graph:

From points (0,80) and (2,20); and (2,20) and (4,5)

• After 2 hours, the amount of medicine has been reduced by a factor of 1/4.

Therefore, after 1 hour, the amount of medicine is reduced by a factor of 1/2.

The completed table is attached below:

Part C

An exponential function is written in the form:

`f(x)=A_o(r)^t\text{ where }\begin{cases}A_o=\text{Starting Value} \\ r=\text{Rate of Decrease}\end{cases}`

Since the amount of medicine is halved every 1 hour:

The rate of decrease = 1/2

The starting amount, Ao = 80 mg

Therefore, an equation that defines f is:

`f(t)=80((1)/(2))^t`

Part D

After 7 hours, when t=7

`\begin{gathered} f(t)=80((1)/(2))^t \\ \implies f(7)=80((1)/(2))^7=0.625\; mg \end{gathered}`

After 7 hours, 0.625 mg of medicine remains in the person's body.