1) To find the length and width of the rectangle, we need to set up a system of equations based on the given information.
Let's assume the length of the rectangle is "L" and the width is "W".
From the problem, we know that the area of the rectangle is 297 square inches, so we can write the equation:
L * W = 297
We are also given that the width is three inches less than four times the length, which can be written as:
W = 4L - 3
Now we have a system of two equations with two variables. To solve, we can substitute the second equation into the first equation:
L * (4L - 3) = 297
Simplify the equation and solve for L:
4L^2 - 3L - 297 = 0
Using factoring or the quadratic formula, we can find the values of L. Once we have the value of L, we can substitute it back into the second equation to find the value of W.
2) To set up the equation for the side lengths of the right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's assume x represents the length of the shorter leg.
According to the given information, the longer leg is 1 more than the shorter leg, so we can represent it as (x + 1).
The hypotenuse has a length of 7, so we can represent it as 7.
Using the Pythagorean theorem, we can set up the equation:
x^2 + (x + 1)^2 = 7^2
Simplifying and expanding the equation will give us the equation needed to solve for the side lengths.
3) To find the consecutive negative odd integers, we can set up an equation based on the given information.
Let's assume the first consecutive negative odd integer is "n". Since the next consecutive odd integer follows n, we can represent it as (n + 2).
The problem states that the product of these two integers is 575, so we can set up the equation:
n * (n + 2) = 575
Solving this equation will give us the values of the consecutive negative odd integers.
4) To find the uniform width of the frame, we need to set up an equation based on the given information.
The canvas picture measures 13 inches by 7 inches, so the area of the canvas is 13 * 7 = 91 square inches.
Let's assume the uniform width of the frame is "w".
The area covered by both the canvas and the frame is 160 square inches, so we can set up the equation:
(13 + 2w)(7 + 2w) = 160
After expanding and simplifying the equation, you can solve for the value of the uniform width "w".
5) To find the dimensions of the rectangle, we can set up equations based on the given information.
Let's assume the length of the rectangle is "L" and the width is "W".
According to the problem, the perimeter of the rectangle is 30 feet, so we can write the equation:
2L + 2W = 30
The area of the rectangle is given as 54 square feet, so we can also write the equation:
L * W = 54
Now we have a system of two equations with two variables. Solving this system will give us the dimensions of the rectangle, assuming the width is smaller than the length.