Use a system of equations to solve the problem.
A rectangular picture frame has a perimeter of 2,200 centimeters and a width that is 300 centimeters less than its length. Find the area of the picture.
_____ cm2
2 L + 2 W = 2200
so
L + W = 1100
W = L - 300
L + (L-300) = 1100
2L - 300 = 1100
2 L = 1400
L = 700
then
W = 400
and
A = L * W = 280,000 cm^2
thank you, big help
Let's represent the length of the picture frame as 'L' and its width as 'W'.
According to the problem, the perimeter of the picture frame is 2,200 centimeters. The formula for the perimeter of a rectangle is P = 2L + 2W.
So we can write the first equation as:
2L + 2W = 2200
The problem also states that the width is 300 centimeters less than the length, meaning W = L - 300.
Now we can substitute the second equation into the first equation:
2L + 2(L - 300) = 2200
Simplifying the equation:
2L + 2L - 600 = 2200
4L - 600 = 2200
Adding 600 to both sides of the equation:
4L = 2800
Dividing both sides by 4:
L = 700
Now we can substitute the value of L back into one of the equations to find the value of W:
W = L - 300
W = 700 - 300
W = 400
The length of the picture frame is 700 centimeters and the width is 400 centimeters.
To find the area of the picture, we can multiply the length by the width:
Area = L * W
Area = 700 * 400
Area = 280,000 square centimeters.
So, the area of the picture is 280,000 cm^2.
To solve this problem using a system of equations, we need to set up two equations based on the given information.
Let's assume that the length of the picture frame is "x" centimeters. Then, the width would be "x - 300" centimeters, since it is 300 centimeters less than the length.
1. Equation for the perimeter:
The perimeter of a rectangle is calculated by adding the lengths of all its sides. In this case, the perimeter is given as 2,200 centimeters. Therefore, we can write the following equation:
2 * (length + width) = perimeter
2 * (x + (x - 300)) = 2200
2. Equation for the area:
The area of a rectangle is found by multiplying its length and width. So, we can write the following equation:
Area = length * width
Area = x * (x - 300)
Now, we have a system of equations:
2 * (x + (x - 300)) = 2200
Area = x * (x - 300)
We can solve this system of equations to find the value of "x" (length) and subsequently calculate the area.
Please note that the above equations are set up based on the given problem and may differ if the problem is slightly modified.