Question
This activity requires you to do some research about building a house. As you conduct this research, you will determine the best function to use for some of the steps. You will also have to determine which information will be important to your task and which information is less important or even irrelevant. Step 1 You have purchased land that is 200 ft. wide by 600 ft. long. In your area, the footprint of a house is allowed to take up 13 of the total area of a lot, and you want your house centered on the lot. Given these constraints, what is the largest area you can have for your house? Include the length and width dimensions. Step 2 Determine how much it will cost to build your house. Use the internet to research the costs of building materials and labor in your area per square foot. You determined the area of the ground floor in Step 1, but will your house have a second floor? Or a finished basement? The number of bedrooms and bathrooms can also increase the building costs. Create a function to model the total costs for your house. Then create a graph to model the function. Step 3 Once your house is built, how much will you pay for it? Many people have to take out a mortgage, which earns compound interest over the time of the loan. Research home loans online to find out the current interest rates and length of loans. Determine how many times the interest will be compounded. What will your monthly mortgage payment be? What type of function is this?
Answer each question in 2-3 sentences
Answer each question in 2-3 sentences
Answers
GPT-4o mini
### Step 1: Determining the Largest Area for the House
To find the largest area allowed for the house on a 200 ft. by 600 ft. lot, we first calculate the total area of the lot, which is \(200 \times 600 = 120,000\) square feet. Since the footprint of the house can take up 13% of the total area, the maximum area for the house can be calculated as \(0.13 \times 120,000 = 15,600\) square feet. To center the house, we could choose dimensions such as 120 ft. wide by 130 ft. long (or any other combination that multiplies to 15,600 sq. ft.), ensuring it fits within the lot's width of 200 ft. and length of 600 ft.
### Step 2: Cost Estimation and Function Creation
To estimate the cost to build the house, I researched that construction costs in my area are approximately $150 per square foot. Given the floor area of 15,600 sq. ft., the total cost can be modeled as \(C(x) = 150x\), where \(x\) is the square footage of the house. The cost will be $2,340,000 for the first floor alone, and additional costs for a second floor or finished basement would increase this amount. A graph of this function would be a straight line representing different total costs for varying house sizes.
### Step 3: Mortgage Calculation
To calculate the monthly mortgage payment, I found that current mortgage interest rates are about 3.5% for a 30-year fixed loan, compounded monthly. Using the formula for a fixed-rate mortgage, \(M = P \frac{r(1+r)^n}{(1+r)^n-1}\), where \(M\) is the monthly payment, \(P\) is the loan amount (total house cost), \(r\) is the monthly interest rate (annual rate / 12), and \(n\) is the number of payments (30 years × 12 months). This results in a monthly payment of approximately $10,472 based on a total loan amount of $2,340,000, representing a geometric function due to the compounding interest.
To find the largest area allowed for the house on a 200 ft. by 600 ft. lot, we first calculate the total area of the lot, which is \(200 \times 600 = 120,000\) square feet. Since the footprint of the house can take up 13% of the total area, the maximum area for the house can be calculated as \(0.13 \times 120,000 = 15,600\) square feet. To center the house, we could choose dimensions such as 120 ft. wide by 130 ft. long (or any other combination that multiplies to 15,600 sq. ft.), ensuring it fits within the lot's width of 200 ft. and length of 600 ft.
### Step 2: Cost Estimation and Function Creation
To estimate the cost to build the house, I researched that construction costs in my area are approximately $150 per square foot. Given the floor area of 15,600 sq. ft., the total cost can be modeled as \(C(x) = 150x\), where \(x\) is the square footage of the house. The cost will be $2,340,000 for the first floor alone, and additional costs for a second floor or finished basement would increase this amount. A graph of this function would be a straight line representing different total costs for varying house sizes.
### Step 3: Mortgage Calculation
To calculate the monthly mortgage payment, I found that current mortgage interest rates are about 3.5% for a 30-year fixed loan, compounded monthly. Using the formula for a fixed-rate mortgage, \(M = P \frac{r(1+r)^n}{(1+r)^n-1}\), where \(M\) is the monthly payment, \(P\) is the loan amount (total house cost), \(r\) is the monthly interest rate (annual rate / 12), and \(n\) is the number of payments (30 years × 12 months). This results in a monthly payment of approximately $10,472 based on a total loan amount of $2,340,000, representing a geometric function due to the compounding interest.