Writing and Using Explicit Rules for Arithmetic Sequences

a

First, let's write the ordered pairs using the numbers we are given. The first number,

281000,

represents the weight of the box as one item is taken out of it. Thus, the ordered pair is

(1, 281000) .

We can write the rest of them in a similar manner.

(1, 281000), (2, 270000), (3, 259000), (4, 248000)

Let's now use the slope-intercept form of the equation to write the function.

y = m x + b

We can calculate the slope

m

by substituting two arbitrary points from above into the Slope Formula. Let's use

(1, 281000)

and

(2, 270000) .

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePointsSubstitute

(1, 281000)

&

(2, 270000)

m = \frac{2 7 0 0 0 0 - 2 8 1 0 0 0}{2 - 1}

Simplify

SubTermsSubtract terms

m = \frac{- 1 1 0 0 0}{1}

DivByOne

\frac{a}{1} = a

m = - 11000

Thus, the slope is

- 11000 .

Now we can find

b

by substituting the slope and a point, let's use

(2, 270000),

into the equation.

y = m x + b

Substitute

m = - 11000

y = (- 11000) x + b

SubstituteII

x = 2

,

y = 270000

270 = - 11 (2) + b

Simplify

MultiplyMultiply

270000 = - 22000 + b

AddEqn

LHS + 22000 = RHS + 22000

292000 = b

RearrangeEqnRearrange equation

b = 292000

Now we can write the function.

y = - 11 x + 292000

b

We know the weights of the box after taking out each item. Hence, if we calculate the differences between consecutive weights, we will know the weight of each item. $281000 - 270000 270000 - 259000 259000 - 248000 = 11000 = 11000 = 11000$ The weight of each yearbook is $11000$ ounces.

c

It is given that the weight of the box when full is $292000$ ounces, and when it's empty is $17000$ ounces. Therefore, by subtracting $17000$ from $292000$ we can calculate the weight of all of the items, not including the weight of the box. $292000 - 17000 = 275000 ounces$ Now if we divide this weight by the weight of $1$ book, we will know how many items were in the box. $\frac{2 7 2 0 0 0}{1 1 0 0 0} = 25 items$ There were $25$ items in the box.

Writing and Using Explicit Rules for Arithmetic Sequences 1.14 - Solution