QuestionBob bought a house for \ $42,000, now worth \$67,500. Find$ m $and$ b $in$ V=m t+b $for$ 0 \leq t \leq 15$.

Studdy Solution

STEP 1

Assumptions1. The initial purchase price of the house is \ $42,000. The appraised value of the house8 years later is \$67,5003. The value of the house over time can be represented by a linear equation$ V=mt+b $, where$ V $is the value of the house,$ m $is the slope of the line (rate of appreciation per year),$ t $is the time in years, and$ b$ is the y-intercept (initial value of the house)
4. The time period considered is0 to15 years

STEP 2

First, we need to find the slope $m$ of the line, which represents the rate of appreciation per year. The slope can be calculated using the formula $m = \frac{V_{2} - V_{1}}{t_{2} - t_{1}}$ where $V_{2}$ and $V_{1}$ are the values of the house at times $t_{2}$ and $t_{1}$ respectively.

STEP 3

Now, plug in the given values for $V_{2}$ , $V_{1}$ , $t_{2}$ , and $t_{1}$ to calculate the slope $m$ .
$m = \frac{\$67,500 - \$42,000}{8 -0}$

STEP 4

Calculate the slope $m$ .
$m = \frac{\$67,500 - \$42,000}{8} = \$3,187.50$

STEP 5

Now that we have the slope $m$ , we can find the y-intercept $b$ , which represents the initial value of the house. Since the house was purchased for \ $42,000,$ b$ is \$42,000.
$b = \$42,000$

STEP 6

Now we have the values for $m$ and $b$ , we can write the linear equation representing the value of the house over time.
$V = mt + b$

STEP 7

Substitute the values of $m$ and $b$ into the equation.
$V = \$3,187.50t + \$42,000$ So, the linear equation representing the value of the house over time is $V = \$3,187.50t + \$42,000$ .

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