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QuestionBob bought a house for \$ 42,000. Now it's worth \$ 67,500.
(a) Find mm and bb in V=mt+bV=m t+b for 0≤t≤150 \leq t \leq 15. (b) Estimate when the house will be worth \$ 72,500.

Studdy Solution

STEP 1

Assumptions1. The initial value of the house is 42,000.Thevalueofthehouse8yearslateris42,000. The value of the house8 years later is 67,5003. The value of the house changes linearly over time4. The linear equation representing the value of the house is V=mt+bV=mt+b, where VV is the value of the house, mm is the slope of the line (rate of change of the house's value), tt is the time in years, and bb is the y-intercept (initial value of the house)

STEP 2

First, we need to find the slope mm of the line. The slope is the rate of change of the house's value over time. It can be found by subtracting the initial value of the house from its value8 years later, and then dividing by the time difference.
m=V8 years−Vinitialt8 years−tinitialm = \frac{V_{8\, years} - V_{initial}}{t_{8\, years} - t_{initial}}

STEP 3

Now, plug in the given values for the initial value of the house, its value8 years later, and the time difference to calculate the slope.
m=$67,500−$42,0008 years−0 yearsm = \frac{\$67,500 - \$42,000}{8\, years -0\, years}

STEP 4

Calculate the slope.
m=$67,500−$42,0008=$3,187. per yearm = \frac{\$67,500 - \$42,000}{8} = \$3,187.\, per\, year

STEP 5

The y-intercept bb is the initial value of the house, which is given as $42,000.
b=$42,000b = \$42,000

STEP 6

Now that we have the slope mm and the y-intercept bb, we can write the linear equation representing the value of the house.
V=mt+bV = mt + b

STEP 7

Plug in the values for mm and bb to write the equation.
V=$3,187.5t+$42,000V = \$3,187.5t + \$42,000

STEP 8

To estimate in how many years after purchase the house will be worth 72,500,wecanset72,500, we can set Vto to 72,500 in the equation and solve for tt.
$72,500=$3,187.5t+$42,000\$72,500 = \$3,187.5t + \$42,000

STEP 9

Subtract 42,000frombothsidesoftheequationtoisolatethetermwith42,000 from both sides of the equation to isolate the term with t$.
$72,500−$42,000=$3,187.5t\$72,500 - \$42,000 = \$3,187.5t

STEP 10

Divide both sides of the equation by 3,187.5tosolvefor3,187.5 to solve for t$.
t=$72,500−$42,000$3,187.5t = \frac{\$72,500 - \$42,000}{\$3,187.5}

STEP 11

Calculate the value of tt.
t=$72,500−$42,000$3,187.5≈9.6 yearst = \frac{\$72,500 - \$42,000}{\$3,187.5} \approx9.6\, yearsSo, the house will be worth $72,500 approximately9.6 years after it was purchased.

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