Question

(from unit 5, lesson 4)

1. the height of water in a bathtub, ( w ), is a function of time, ( t ). let ( p ) represent this function. height is measured in inches and time in minutes. match each statement in function notation with a description.

function notation	description
b. ( p(4) = 10 )	ii. the bathtub starts out with no water.
c. ( p(10) = 4 )	iii. after 10 minutes, the height of the water is 4 inches.
d. ( p(20) = 0 )	iv. the height of the water is 10 inches after 4 minutes.

(from unit 5, lesson 4)

1. the graph of function ( f ) gives the distance of a dog from a post, in feet, as a function of time, in seconds, since its owner left.

find the approximate values of ( f(20) ) and of ( f(140) ).

Explanation:

Question 7 (Matching Function Notation to Descriptions)

Step 1: Analyze \( P(0) = 0 \)

The function \( P(t) \) represents the height of water at time \( t \) minutes. When \( t = 0 \) (the start time), \( P(0) = 0 \) means the height of water is 0 inches at the beginning. This matches description ii: "The bathtub starts out with no water."

Step 2: Analyze \( P(4) = 10 \)

Here, \( t = 4 \) minutes and \( P(4) = 10 \) inches. So after 4 minutes, the height of the water is 10 inches. This matches description iv: "The height of the water is 10 inches after 4 minutes."

Step 3: Analyze \( P(10) = 4 \)

For \( t = 10 \) minutes, \( P(10) = 4 \) inches. This means after 10 minutes, the height of the water is 4 inches. This matches description iii: "After 10 minutes, the height of the water is 4 inches."

Step 4: Analyze \( P(20) = 0 \)

When \( t = 20 \) minutes, \( P(20) = 0 \) inches. So after 20 minutes, the height of the water is 0 inches (the tub is empty). This matches description i: "After 20 minutes, the bathtub is empty."

To find \( f(20) \) and \( f(140) \), we analyze the graph of the function \( f \), where the x - axis represents time (in seconds) and the y - axis represents the distance of the dog from the post (in feet).

Step 1: Find \( f(20) \)

We look at the x - value \( t = 20 \) seconds on the graph. By examining the graph, we find the corresponding y - value (distance from the post) at \( t = 20 \). From the graph, when \( t = 20 \) seconds, the distance of the dog from the post is approximately 3 feet. So \( f(20)\approx3 \).

Step 2: Find \( f(140) \)

Next, we look at the x - value \( t = 140 \) seconds. By looking at the graph, we find the corresponding y - value at \( t = 140 \). From the graph, when \( t = 140 \) seconds, the distance of the dog from the post is approximately 2 feet. So \( f(140)\approx2 \).

Answer:

a. \( P(0) = 0 \) matches ii.
b. \( P(4) = 10 \) matches iv.
c. \( P(10) = 4 \) matches iii.
d. \( P(20) = 0 \) matches i.

Question 8 (Finding \( f(20) \) and \( f(140) \) from the Graph)