Austin bought a new house. The value of his house is modeled by the function H\left( x \right) = 120000{\left( {1.20} \right)^{\left( {\frac{1}{2}x} \right)}} where x is the number of years since he purchased the house. Looking at the model by what approximate percentage rate is the value of his house increasing? HHHHHHHEEEEEEELLLLLLPPPPP!!!!!!!!!
Sorry - not much TeX here. It appears that you mean
H(x) = 120000*1.20^(1/2 x)
That means that H grows by 20% every 2 years. But, if we want an annual rate, then we have to realize that this is the same as
H(x) = 120000*(√1.2)^x
√1.2 = 1.095
So, we wind up with
H(x) = 120000*1.095^x
So, the annual appreciation is about 9.5%
To find the approximate percentage rate at which the value of Austin's house is increasing, we can use the concept of the derivative. The derivative of a function measures its rate of change.
In this case, the value of Austin's house is represented by the function H(x) = 120,000 * (1.20)^(1/2x). To calculate the derivative of this function, we can use the chain rule.
Step 1: Apply the power rule for differentiation. The derivative of (1.20)^(1/2x) with respect to x is equal to (1/2x) * (1.20)^(1/2x - 1) * ln(1.20).
Step 2: Multiply the derivative by the constant term 120,000 to find the rate at which the value of the house is changing.
Putting it all together, we get:
H'(x) = 120,000 * (1/2x) * (1.20)^(1/2x - 1) * ln(1.20)
Now, to find the percentage rate of change, we can divide the derivative H'(x) by the value of the function H(x) and multiply by 100:
Percentage rate of change = (H'(x) / H(x)) * 100
Substituting the derivative and the function H(x), we can simplify:
Percentage rate of change = ((120,000 * (1/2x) * (1.20)^(1/2x - 1) * ln(1.20)) / (120,000 * (1.20)^(1/2x))) * 100
Simplifying further:
Percentage rate of change = ((1/2x) * (1.20)^(1/2x - 1) * ln(1.20)) / (1.20)^(1/2x) * 100
Now, you can substitute the value of x (the number of years since Austin purchased the house) into the formula to calculate the approximate percentage rate at which the value of his house is increasing.