1.3+Writing+an+Equation

1.3 Writing an Equation
YRK Jan 25 Math 8D

__NOTES__
 * One half of the perimeter (l+w) gives the distance of the width and length.
 * If L is the length, then the width must be half the perimeter minus the length.
 * Width= half the perimeter - length

__Problem 1.3__ The rectangle below has a perimeter of 20 meters and a side of length //L// meters.

A. Express the length of each side of the rectangle in terms of L. That is, write an expression that contains the variable L to represent the length of each side.

X= 6 L= 10 10 - 6= 4 width= 4 length= 6

B. Write an equation for the area, A, of the rectangle in terms of L.

Area = L(10-L)

C. If the length of a side of the rectangle is 6 meters, what is the area?

L(10-L) 6(10-6) 6 x 4 = 24

D. Use a calculator to make a table and a graph for your equation. Show x values from 1 to 10 and y values from 1 to 30. Compare your table and graph to those you made in problem 1.1.

TABLE OF EQUATION L(10-L)


 * = X (length) ||= Y(area) ||
 * = 0 ||= 0 ||
 * = 1 ||= 9 ||
 * = 2 ||= 1 ||
 * = 3 ||= 21 ||
 * = 4 ||= 24 ||
 * = 5 ||= 25 ||
 * = 6 ||= 24 ||
 * = 7 ||= 21 ||
 * = 8 ||= 16 ||
 * = 9 ||= 9 ||
 * = 10 ||= 0 ||

GRAPH FOR EQUATION L(10-L)


 * the graph in problem 1.1 has no difference compared to this graph

__FOLLOW-UP 1.3__ //consider rectangles with a perimeter of 60 meters.// 1a. As in problem 1.3, draw a rectangle to represent this situation. Label on side l, and label the other sides in terms of l.

l 30-l

1b. Write an equation for the area, A, in terms of L. A= L(30-L)

1c. Make a table for your equation. Then use your table to estimate the greatest area possible for a rectangle with a perimeter of 60 meters. Give the side lengths of this rectangle.

The greatest possible area = 225m^2 Side lengths= 15m

1d.

1e. How can you use your graph to find the maximum area? How does your graph show the side length that corresponds to the maximum area?

You have to look at the maximum value of the parabola. In this case (maximum area= 225m^2), the side lengths are on the x- axis which is on 15m, since the greatest area is a square so, 15 x 15 = 225

//An equation for the area of rectangles with a certain fixed perimeter is// A = l(35-l). 2a. Draw a rectangle to represent this situation. Label one side l, and label the other sides in terms of l. An equation for the area with a certain fixed perimeter is A= l(35-l)

l 35-l perimeter = 70

2b. If the length of a side of a rectangle with this fixed perimeter is 20 meters, what is the area?

If the length of a side of a rectangle with this fixed perimeter is 20m, what is the area? The area is: A= L(35-L) A= 20(35-20) A= 20(15) A= 300m^2

2c. Describe two ways you could find th perimeter for the rectangles represented by this equation. What is the perimeter?

One way you could find the perimeter is by using the equations and multiply by 2. Or you could follow the equation and what ever two multiples you end up with multiply each multiples by 2 and then add them up. The perimeter is 70.

2d. Describe the graph of this equation.

2e. What is the maximum area for this family of rectangles? What dimensions correspond to this maximum area? Explain how you found your answers.

Squares are the largest in Area in the family of rectangles. The same dimensions all around.

3.How can you write an equation for the area in terms of the length of a side?

If I know the perimeter, then I divide that by half, because l+w is half of the other l+w and then you subtract the half to the length of side, giving you the l x w, which then you multiply to find the area.

4.Graphs of quadratic functions are called parabolas. Describe the characteristics of the parabolas you have seen so far. They are curved up of curved down in a "U" shape having both maximum and minimum values.

5. Study the graphs, tables and equations for areas of rectangles with fixed perimeters. Which representation is most useful for predicting the maximum area?

On the graphs, the maximum point is the most useful and on a table the maximum value, or middle number is the most useful to find the maximum area.