2 exploration: recognizing inverse variation
go to bigideasmath.com for an interactive tool to investigate this exploration.
work with a partner. the table shows the length x (in inches) and the width y (in inches) of a rectangle. the area of each rectangle is 64 square inches.

a. complete the table.

b. describe the relationship between x and y. explain why y is said to vary inversely with x.

c. draw a scatter plot of the data. what are the characteristics of the graph?

d. write an equation that represents y as a function of x.

communicate your answer
3. how can you recognize when two quantities vary directly or inversely?

4. does the flapping rate of the wings of a bird vary directly or inversely with the length of its wings? explain your reasoning.

(table and graph as shown in the image: table with x values 1,2,4,8,16,32,64 and empty y column; scatter plot grid with y-axis 0-60 and x-axis 0-40 (approx))

Question

2 exploration: recognizing inverse variation
go to bigideasmath.com for an interactive tool to investigate this exploration.
work with a partner. the table shows the length x (in inches) and the width y (in inches) of a rectangle. the area of each rectangle is 64 square inches.

a. complete the table.

b. describe the relationship between x and y. explain why y is said to vary inversely with x.

c. draw a scatter plot of the data. what are the characteristics of the graph?

d. write an equation that represents y as a function of x.

communicate your answer
3. how can you recognize when two quantities vary directly or inversely?

4. does the flapping rate of the wings of a bird vary directly or inversely with the length of its wings? explain your reasoning.

(table and graph as shown in the image: table with x values 1,2,4,8,16,32,64 and empty y column; scatter plot grid with y-axis 0-60 and x-axis 0-40 (approx))

Accepted Answer

### Part (a)
# Explanation:
## Step1: Recall the area formula for a rectangle
The area $ A $ of a rectangle is given by $ A = x \times y $. We know that $ A = 64 $ square inches, so we can solve for $ y $ in terms of $ x $: $ y=\frac{64}{x} $.

## Step2: Calculate $ y $ for each $ x $
- When $ x = 1 $: $ y=\frac{64}{1}=64 $
- When $ x = 2 $: $ y=\frac{64}{2}=32 $
- When $ x = 4 $: $ y=\frac{64}{4}=16 $
- When $ x = 8 $: $ y=\frac{64}{8}=8 $
- When $ x = 16 $: $ y=\frac{64}{16}=4 $
- When $ x = 32 $: $ y=\frac{64}{32}=2 $
- When $ x = 64 $: $ y=\frac{64}{64}=1 $

# Answer:
| $ x $ | $ y $ |
|--------|--------|
| 1      | 64     |
| 2      | 32     |
| 4      | 16     |
| 8      | 8      |
| 16     | 4      |
| 32     | 2      |
| 64     | 1      |

### Part (b)
# Explanation:
## Step1: Analyze the relationship
As $ x $ (length) increases, $ y $ (width) decreases. We can check the product $ x \times y $ for each pair: $ 1\times64 = 64 $, $ 2\times32 = 64 $, $ 4\times16 = 64 $, etc. The product is constant ($ 64 $).

## Step2: Define inverse variation
Two quantities $ x $ and $ y $ vary inversely if their product $ xy = k $ (where $ k $ is a non - zero constant). Here, $ xy = 64 $ (constant), so $ y $ varies inversely with $ x $.

# Answer:
The relationship between $ x $ and $ y $ is that as $ x $ increases, $ y $ decreases. $ y $ varies inversely with $ x $ because the product $ xy=64 $ (a constant), which fits the definition of inverse variation ($ xy = k $, $ k
eq0 $).

### Part (c)
# Explanation:
## Step1: Plot the points
The data points are $(1, 64)$, $(2, 32)$, $(4, 16)$, $(8, 8)$, $(16, 4)$, $(32, 2)$, $(64, 1)$. When we plot these points on a scatter plot (with $ x $ on the horizontal axis and $ y $ on the vertical axis), we observe the following characteristics:

## Step2: Describe the graph
The graph is a curve that approaches the $ x $ - axis as $ x $ increases and approaches the $ y $ - axis as $ x $ approaches $ 0 $ (from the positive side). It is the graph of a hyperbola (since $ y=\frac{64}{x} $ is a rational function whose graph is a hyperbola), and it is in the first quadrant (since $ x $ and $ y $ represent length and width, so they are positive).

# Answer:
When we plot the points $(1, 64)$, $(2, 32)$, $(4, 16)$, $(8, 8)$, $(16, 4)$, $(32, 2)$, $(64, 1)$ on a scatter plot:
- The graph is a curve (not a straight line).
- As $ x $ increases, $ y $ decreases, and the curve gets closer to the $ x $ - axis (as $ x
ightarrow\infty $) and closer to the $ y $ - axis (as $ x
ightarrow0^{+} $).
- All the points lie in the first quadrant.

### Part (d)
# Explanation:
## Step1: Start with the area formula
We know that the area of the rectangle $ A=x\times y $ and $ A = 64 $.

## Step2: Solve for $ y $ in terms of $ x $
From $ 64=x\times y $, we can rearrange the formula to get $ y=\frac{64}{x} $ (where $ x>0 $ since $ x $ represents the length of a rectangle).

# Answer:
The equation that represents $ y $ as a function of $ x $ is $ y=\frac{64}{x} $ (or $ y = 64x^{-1} $).

### Question 3
# Explanation:
## Step1: Define direct variation
Two quantities $ x $ and $ y $ vary directly if $ y=kx $ (or $ \frac{y}{x}=k $), where $ k $ is a non - zero constant. In this case, as $ x $ increases, $ y $ increases (and as $ x $ decreases, $ y $ decreases) and the ratio of $ y $ to $ x $ is constant.

## Step2: Define inverse variation
Two quantities $ x $ and $ y $ vary inversely if $ xy = k $ (or $ y=\frac{k}{x} $), where $ k $ is a non - zero constant. In this case, as $ x $ increases, $ y $ decreases (and as $ x $ decreases, $ y $ increases) and the product of $ x $ and $ y $ is constant.

# Answer:
To recognize direct variation between two quantities $ x $ and $ y $: Check if $ \frac{y}{x}=k $ (a constant) or if $ y = kx $ for some non - zero constant $ k $. As $ x $ increases, $ y $ increases (and vice - versa) in direct variation.

To recognize inverse variation between two quantities $ x $ and $ y $: Check if $ xy=k $ (a constant) or if $ y=\frac{k}{x} $ for some non - zero constant $ k $. As $ x $ increases, $ y $ decreases (and vice - versa) in inverse variation.

### Question 4
# Explanation:
## Step1: Analyze the relationship
A bird with longer wings does not need to flap its wings as frequently (lower flapping rate) to achieve a certain level of lift or movement, and a bird with shorter wings needs to flap its wings more frequently (higher flapping rate) to achieve the same.

## Step2: Relate to variation types
Let $ f $ be the flapping rate and $ l $ be the wing length. As $ l $ (wing length) increases, $ f $ (flapping rate) decreases. We can think of the "work" or "effect" of flapping (related to the product of flapping rate and wing length) as approximately constant (to achieve a certain flight effect). So, if we consider the relationship, as $ l $ increases, $ f $ decreases, and the product $ f\times l $ is approximately constant (for a given flight requirement).

# Answer:
The flapping rate of the wings of a bird varies inversely with the length of its wings. Reason: Generally, longer - winged birds can cover more distance with each flap, so they don't need to flap as often (lower flapping rate) to move a certain distance or maintain flight. Shorter - winged birds need to flap more often (higher flapping rate) to achieve the same. So, as wing length (let's say $ l $) increases, flapping rate (let's say $ f $) decreases, and the product $ f\times l $ is approximately constant (for a given flight task), which is consistent with inverse variation ($ fl = k $, $ k$ is a constant related to flight requirements).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Recall the area formula for a rectangle

Step2: Calculate \( y \) for each \( x \)

Step1: Analyze the relationship

Step2: Define inverse variation

Step1: Plot the points

Step2: Describe the graph

Answer:

Part (b)