Q45. - Algebra 2

Question

45. BUSINESS Refer to the signs below. At what distance do the two stores charge the same amount for a balloon arrangement

Step-by-Step Solution

Answer

The distance at which the two stores charge the same amount for a balloon arrangement is 10 miles.

Step 1 – State the concept

The slope-intercept form of an equation of straight line is $y=mx+c$, where $m$ is the slope and $c$ is the $y$-intercept.

 Geometrically the slope represents the rate of change of $y$ and the $y$-intercept represents the value of $y$ when $x=0$

Step 2 – List the given data

The initial price of balloon arrangement at Conrad’s Balloon Bouquets is $20. The delivery charge is $3 per mile.

 The initial price of balloon arrangement at The Balloon House is $30. The delivery charge is $2 per mile.

Step 3 – Write the equations

Let $x$ be the distance in miles that the balloon arrangement has to be delivered to and $y$ be the total price.

 It is given that the initial price at Conrad’s Balloon Bouquets, obviously represented by $x=0$, is $20.

So, the value of $y$ when $x=0$ is 20. That is, the $y$-intercept is 20.

 Then, $c=20$.

 It is given that the price will increase by $3 per mile for delivery.

Thus, the rate of change of$y$ is 3. So, the slope is 3.

 Then, $m=3$.

 Put $m=3$ and $c=20$ in $y=mx+c$ to get,

$y=3x+20$

 So, the equation that represents the total price of a balloon arrangement at Conrad’s Balloon Bouquets for $x$ miles of delivery is $y=3x+20$.

 Similarly, it is given that the initial price at The Balloon House, obviously represented by $x=0$, is $30.

So, the value of $y$ when $x=0$ is 30. That is, the $y$-intercept is 30.

 Then, $c=30$.

 It is given that the price will increase by $2 per mile for delivery.

Thus, the rate of change of $y$ is 2. So, the slope is 2.

 Then, $m=2$.

 Put $m=2$ and $c=30$ in $y=mx+c$ to get,

$y=2x+30$

 So, the equation that represents the total price of a balloon arrangement at The Balloon House for $x$ miles of delivery is $y=2x+30$

Step 4 – Compute the required distance

The objective is to find the distance at which the two stores charge the same amount for a balloon arrangement.

 Let such a distance be $p$ miles.

 Put $x=p$ in $y=3x+20$ to get,

$y=3p+20$

 This is the price at Conrad’s Balloon Bouquets.

 Put $x=p$ in $y=2x+30$ to get,

$y=2p+30$

 This is the price at The Balloon House.

 According to the assumption, these prices are equal.

 Then,

 $3p+20=2p+30$

 

$3p+20-2p=2p+30-2p$  (Subtract $2p$ from both sides)

 $p+20=30$   (Simplify)

 $p+20-20=30-20$   (Subtract 20 from both sides)

 $p=10$   (Simplify)

 This implies that the distance at which the two stores charge the same amount for a balloon arrangement is 10 miles.