FC: Quadratic Word Problems Question 13 | By: Chris Kiley
1: Problem and Goals of the Word Problem | A rectangular field is 300 yards by 500 yards. A roadway of width "x" yards is to be built inside the field. This problem concerns the region inside the roadway. | 1. Write the length and width of the region as functions of "x". What kind of functions are these? 2. Write the area of the region as a function of "x". What kind of function is this? 3. Predict the area of the region if "x = 5,10,and 15" 4. What is the widest the roadway can be and still leave 100,000 square yards in the region? Explain. 5. Find the value of "x" which makes the roadway have an area equal to the area of the field.
2: Question 1 | The function of length is f(x)=500-2x The function of width is f(x)=300-2x These two functions are both linear functions
3: Question 2 | The area of the function is (500-2x)(300-2x) This is a quadratic function | Area is length x width. Therefore the bigger rectangles area is 500x300=150,000 square yds. To find the inner rectangles area you must subtract the inner from the larger rectangle, as shown in the equation at the top.
4: Question 3 | If x was 5 the area would be 142,100 square yds. If x was 10 the area would be 134,400 square yds. If x was 15 the area would be 126,900 square yds. To do this you take the area equation of (500-2x)(300-2x) and replace the x values with the given value. Then you find the area in square yds if x was equal to that value.
5: Question 4 | The widest that the roadway can be with still leaving 100,000 square yards in the region is 100 yards. By applying this width of 100 yds with the length of 500 yds, you get an area of 50,000 square yds. Then the remaining area of the region would be 100,000 square yds. When adding 100,000 and 50,000 you get 150,000 square yds which is the original area of the rectangle.
6: Question 5 | The value of x which makes the roadway have an area equal to that of the field is 150. With x being 150, you multiply its length of 500 by its width of 150 which makes the roadway's area 75,000 square yds. Since the rectangle has an area of 150,000 square yds and the roadway has an area of 75,000 square yds, then the rest of the field must have an area of 75,000 square yds as well, which makes them equal.