Question

example 7

1. geometry the length of a leg of an isosceles right triangle compared to its area is provided in the table.

leg length (in) | area (in²)
0 | 0
2 | 1
4 | 8
6 | 18
a. determine linearity.
b. graph.

1. temperature the temperature in degrees celsius compared to the equivalent temperature in degrees fahrenheit is provided in the table.

temperature (°c) | temperature (°f)
-20 | -4
0 | 32
10 | 50
20 | 68
40 | 104
a. determine linearity.
b. graph.

Explanation:

20a: Determine Linearity

Step1: Recall linearity test (constant slope)

For a linear relationship \( y = mx + b \), the slope \( m=\frac{\Delta y}{\Delta x} \) should be constant. Let \( x \) = Temperature (°C), \( y \) = Temperature (°F).

Step2: Calculate slope between first two points

\( x_1=-20, y_1=-4 \); \( x_2=0, y_2=32 \).
Slope \( m_1=\frac{32 - (-4)}{0 - (-20)}=\frac{36}{20}=1.8 \).

Step3: Calculate slope between next two points

\( x_2=0, y_2=32 \); \( x_3=10, y_3=50 \).
Slope \( m_2=\frac{50 - 32}{10 - 0}=\frac{18}{10}=1.8 \).

Step4: Calculate slope between next two points

\( x_3=10, y_3=50 \); \( x_4=20, y_4=68 \).
Slope \( m_3=\frac{68 - 50}{20 - 10}=\frac{18}{10}=1.8 \).

Step5: Calculate slope between last two points

\( x_4=20, y_4=68 \); \( x_5=40, y_5=104 \).
Slope \( m_4=\frac{104 - 68}{40 - 20}=\frac{36}{20}=1.8 \).

Step1: Identify axes

• X - axis: Temperature (°C) (values: -20, 0, 10, 20, 40).
• Y - axis: Temperature (°F) (values: -4, 32, 50, 68, 104).

Step2: Plot points

• (-20, -4), (0, 32), (10, 50), (20, 68), (40, 104).

Step3: Draw line

Connect the points; they lie on a straight line (consistent with linearity).

Step1: Recall linearity test (constant slope)

Let \( x \) = Leg Length (in), \( y \) = Area (in²). Points: (0, 0), (2, 1), (4, 8), (6, 18).

Step2: Calculate slope between first two points

\( m_1=\frac{1 - 0}{2 - 0}=\frac{1}{2}=0.5 \).

Step3: Calculate slope between next two points

\( m_2=\frac{8 - 1}{4 - 2}=\frac{7}{2}=3.5 \).
Since \( m_1
eq m_2 \), slope is not constant.

Answer:

The relationship is linear (constant slope \( 1.8 \)).

20b: Graph