Question

Paula knits hats. The number of hats h she can knit varies directly with the amount of yarn (y) in yards she has. The constant of proportionality is 42. Write an equation to represent the relationship between the numbers of yards of yarn Paula has and the number of hats she can knit. h=
What does the constant if proportionality 42, represent in this problem?

Complete the table showing the yards of yarn and the number of hats Paula can knit.

Answer 1

Answer:

Part 1) The equation is $h=42y$

Part 2) The number of hats Paula can knit is 42 per yard of yarn

Part 3) see the procedure

Part 4) For 378 yards of yarn Paula can knit 9 hats

Step-by-step explanation:

Let

h -----> the number of hats

y ----> the amount of yarn in yards  

k ---> constant of proportionality

Part 1) Write an equation to represent the relationship between the numbers of yards of yarn Paula has and the number of hats she can knit

we know that

A relationship between two variables, h, and y, represent a proportional variation if it can be expressed in the form $h/y=k$ or $h=ky$

we have

$k=42$

substitute

$h=42y$

Part 2) What does the constant if proportionality 42, represent in this problem?

The constant of proportionality represent the slope of the linear equation

The value of the slope is $42\ \frac{hats}{yd}$

That means that the number of hats Paula can knit is 42 per yard of yarn

Part 3) Complete the table showing the yards of yarn and the number of hats Paula can knit

we have

$h=42y$

Part a) For y=0 yd Find the value of h

substitute in the linear equation the value of y and solve for h

$h=42(0)=0\ hats$

Part b) For h=3 hats Find the value of y

substitute in the linear equation the value of h and solve for y

$3=42y$

$y=3/42\ yd$  ----> $y=1/14\ yd$

Part c) For y=210 yd Find the value of h

substitute in the linear equation the value of y and solve for h

$h=42(210)=8,820\ hats$

Part d) For y=378 yd Find the value of h

substitute in the linear equation the value of y and solve for h

$h=42(378)=15,876\ hats$

Part 3) For h=11 hats Find the value of y

substitute in the linear equation the value of h and solve for y

$11=42y$

$y=11/42\ yd$  

Part 4) Interpret the meaning of the point (378,9)

we have that

y=378 yd

h=9 hats

That means -----> For 378 yards of yarn Paula can knit 9 hats