A rectangle has perimeter 64cm and area 23cm squared. Solve the following system of equations to find the rectangle's dimensions.
l=(23/w)
l+w=32
to solve system of equations you solve for one variable and plug into the next
in this case you have an equation for I so plug into the next equation and solve
(23/w) + w = 32
To solve the given system of equations, we can start by substituting the value of "l" from the first equation into the second equation.
Given:
l = 23/w
l + w = 32
Substitute the value of "l" in terms of "w" into the second equation:
23/w + w = 32
Next, multiply both sides of the equation by "w" to eliminate the denominator:
23 + w^2 = 32w
Rearrange the equation to form a quadratic equation:
w^2 - 32w + 23 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -32, and c = 23. Substituting these values into the quadratic formula:
w = (-(-32) ± √((-32)^2 - 4(1)(23))) / (2(1))
w = (32 ± √(1024 - 92)) / 2
w = (32 ± √932) / 2
Now, we simplify this equation:
w = (32 ± √932) / 2
w = (32 ± √(4 × 233)) / 2
w = (32 ± 2√233) / 2
w = 16 ± √233
Therefore, the possible values of "w" are 16 + √233 and 16 - √233.
Now, we can substitute these values back into the first equation to find the corresponding values of "l". Let's calculate both cases:
For w = 16 + √233,
l = 23 / (16 + √233)
l = 23(16 - √233) / (16 + √233)
l ≈ 6.162 cm
For w = 16 - √233,
l = 23 / (16 - √233)
l = 23(16 + √233) / (16 - √233)
l ≈ 37.838 cm
Therefore, the possible dimensions of the rectangle are approximately 6.162 cm × 16 + √233 cm and 37.838 cm × 16 - √233 cm.