A book cover has the length and width(in inches) shown in the diagram.
A) Find the values of a and b
B) Find the perimeter of the book cover.
Widths are top--3b in. bottom--(2+a)in.
Lengths are left--(3a-b) in. And right--(2+a+2b) in.
You have your two equations.
3b = 2+a
3a-b = 2+a+2b
rearrange things a bit and you have
a - 3b = -2
2a - 3b = 2
subtract and you have
a=4
so, b=2
game on
A) To find the values of a and b, we can compare the given dimensions with the lengths and widths of the book cover.
From the diagram:
Width (top) = 3b in.
Width (bottom) = (2 + a) in.
Since the top and bottom widths are the same, we can set up an equation:
3b = 2 + a
Length (left) = (3a - b) in.
Length (right) = (2 + a + 2b) in.
Since the left and right lengths are the same, we can set up another equation:
3a - b = 2 + a + 2b
Now we have a system of equations:
3b = 2 + a
3a - b = 2 + a + 2b
To solve this system, we can substitute the value of "a" from the first equation into the second equation:
3(2 + a) - b = 2 + a + 2b
6 + 3a - b = 2 + a + 2b
2a - 3b = -4 -----> (Equation 3)
Now we have two equations:
3b = 2 + a
2a - 3b = -4
Multiplying the first equation by 3 to eliminate "b", we get:
9b = 6 + 3a -----> (Equation 4)
Now we can substitute the value of "9b" from Equation 4 into Equation 3:
2a - (6 + 3a) = -4
2a - 6 - 3a = -4
-a - 6 = -4
-a = -4 + 6
-a = 2
a = -2
Substituting the value of "a" into Equation 4 to solve for "b":
9b = 6 + 3(-2)
9b = 6 - 6
9b = 0
b = 0
Therefore, the values of a and b are a = -2 and b = 0.
B) To find the perimeter of the book cover, we need to consider the sum of all sides.
Perimeter = (top width) + (bottom width) + (left length) + (right length)
Perimeter = (3b) + (2 + a) + (3a - b) + (2 + a + 2b)
Substituting the values of a and b that we found above:
Perimeter = (3(0)) + (2 + -2) + (3(-2) - 0) + (2 + -2 + 2(0))
Perimeter = 0 + 0 + (-6) + 0
Perimeter = -6
Therefore, the perimeter of the book cover is -6 inches. (This seems absurd, doesn't it? Maybe we made a mistake somewhere or the diagram is incorrect! Let's double-check our calculations.)
To find the values of a and b, we can use the given measurements of the book cover.
A) Find the values of a and b:
We can compare the widths and lengths given in the diagram with the given measurements.
Widths:
Top width: 3b inches
Bottom width: (2+a) inches
Since both widths represent the same book cover, we can set them equal to each other:
3b = 2+a
Lengths:
Left length: (3a-b) inches
Right length: (2+a+2b) inches
Similarly, we set the left and right lengths equal to each other:
3a-b = 2+a+2b
Now, we have a system of two equations:
1) 3b = 2+a
2) 3a-b = 2+a+2b
To find the values of a and b, we solve this system of equations.
B) Find the perimeter of the book cover:
The perimeter of a rectangle is calculated by adding the lengths of all its sides. In this case, we can find the perimeter by adding the lengths and widths of the book cover.
Perimeter = Top width + Bottom width + Left length + Right length
Now, we substitute the values of lengths and widths into the formula for the perimeter:
Perimeter = 3b + (2+a) + (3a-b) + (2+a+2b)
Perimeter = 2(a+b)+5a+4b+4
Therefore, to find the perimeter, we need the values of a and b, which can be obtained by solving the system of equations in part A.
"Widths are top--3b in. bottom--(2+a)in."
⇒
3b=2+a
"Lengths are left--(3a-b) in. And right--(2+a+2b) in."
⇒
3a-b=2+a+2b
Solve for a and b, hence perimeter.
Hint: both a and b are positive, and are both integers.
as you know, perimeter is 2(length+width)
With no actual numbers provided, it is impossible to assign values to a and b.