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Math  /  Algebra

QuestionThe price of a certain object was $12.50\$ 12.50 in 1995 and $15.50\$ 15.50 in 2001. Assume that the price has been increasing at a constant rate since 1990.
Determine the rate of change in the price of the object, that is, how much the price increases each year. Do not round your answer. □\square Write an equation to model the price of the object over time, with yy representing the price of the object in dollars, and xx representing the number of years since 1990. Write your final answer in the form y=mx+by=m x+b. (You may use the point-slope form to help you find the equation.) □\square Looking at your equation, what was the price of the object in the year 1990? If necessary, round to your answer to two decimal places. \ \squareHowmuchdoestheobjectcostnow,in2024?$ How much does the object cost now, in 2024? \$ \square$

Studdy Solution

STEP 1

1. The price increases at a constant rate from 1990 onwards.
2. The price was 12.50in1995and12.50 in 1995 and 15.50 in 2001.
3. We need to find the rate of change, the equation of the price over time, the price in 1990, and the price in 2024.

STEP 2

1. Determine the rate of change in price per year.
2. Write the equation modeling the price over time.
3. Calculate the price of the object in 1990.
4. Calculate the price of the object in 2024.

STEP 3

To find the rate of change, use the formula for the slope m m between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), where y y represents the price and x x represents the number of years since 1990.
The points are (5,12.50)(5, 12.50) for 1995 and (11,15.50)(11, 15.50) for 2001.
m=y2−y1x2−x1=15.50−12.5011−5=36=0.5 m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{15.50 - 12.50}{11 - 5} = \frac{3}{6} = 0.5
The rate of change is 0.50.5 dollars per year.

STEP 4

Use the point-slope form of a line to write the equation: y−y1=m(x−x1) y - y_1 = m(x - x_1) .
Choose the point (5,12.50)(5, 12.50) and use m=0.5 m = 0.5 :
y−12.50=0.5(x−5) y - 12.50 = 0.5(x - 5)
Simplify to get the equation in slope-intercept form y=mx+b y = mx + b :
y−12.50=0.5x−2.5 y - 12.50 = 0.5x - 2.5
y=0.5x+10 y = 0.5x + 10

STEP 5

To find the price in 1990, substitute x=0 x = 0 into the equation y=0.5x+10 y = 0.5x + 10 :
y=0.5(0)+10=10 y = 0.5(0) + 10 = 10
The price in 1990 was \$10.00.

STEP 6

To find the price in 2024, substitute x=34 x = 34 (since 2024 is 34 years after 1990) into the equation y=0.5x+10 y = 0.5x + 10 :
y=0.5(34)+10=17+10=27 y = 0.5(34) + 10 = 17 + 10 = 27
The price in 2024 is \$27.00.
The rate of change is 0.50.5 dollars per year. The equation modeling the price is y=0.5x+10 y = 0.5x + 10 . The price in 1990 was \$10.00. The price in 2024 is \$27.00.

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