Exercise 29 Page 184

Remember that equations written in point-slope form follow a specific format. y- y_1= m(x- x_1) In this form, m is the slope of the line and ( x_1, y_1) is a point on the line. In our case, let x represents the number of chirps and y the temperature. We know that when the temperature is 50^(∘) F a cricket chirps 40 times, this can be rewritten as a ordered pair. (Chirps,Temperature) → (40,50) For each rise in temperature of 0.25 ^(∘) F, the cricket makes and additional chirp each minute. Then, another ordered point can be (41,50.25) which represents the number of chirps when the temperature is 50.25^(∘) F. Now that we have two points, let's determine the slope with the slope formula.

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 40,50) & ( 41,50.25)

m=50.25- 50/41- 40

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Simplify right-hand side

SubTerms

Subtract terms

m=0.25/1

DivByOne

a/1=a

m=0.25

Now that we know the slope is 0.25, we can write the equation of the line in point-slope form. We can use either of the given points as (x_1,y_1) in our equation. Let's use ( 40, 50).

y-y_1=m(x-x_1)

SubstituteValues

Substitute values

y- 50= 0.25(x- 40)

Distr

Distribute 0.25

y-50=0.25x-0.25(40)

Multiply

Multiply

y-50=0.25x-10

AddEqn

LHS+50=RHS+50

y-50+50=0.25x-10+50

AddTerms

Add terms

y=0.25x+40

This equation represents the temperature per number of chirps. We want to find the temperature if we count 100 chirps in 1 minute. To do so, let's substitute x= 100 into the obtained equation.

y=0.25x+40

Substitute

x= 100

y=0.25( 100)+40

Multiply

Multiply

y=25+40

AddTerms

Add terms

y=65

The temperature is 65^(∘) F when we count 100 chirps per minute.