The sum of the length l and the width w of a rectangular area is 220 meters.
(a) Write w as a function of l.
w =
(b) Write the area A as a function of l.
A =
(c) Find the dimensions that produce the greatest area.
l =
w =
a) w = 220 - l
b) A = lw = l(220-l)
c) A square would give you the greatest area.
(a) Since the sum of the length l and the width w is 220 meters, we can write w as a function of l as:
w = 220 - l
(b) The area A of a rectangle is given by A = lw. Using the expression for w from part (a), we can substitute it into the equation to get:
A = l(220 - l)
Simplifying further, we have:
A = 220l - l^2
(c) To find the dimensions that produce the greatest area, we need to find the maximum value of the area function A = 220l - l^2. Since this is a quadratic function, we know that the maximum value occurs at the vertex of the parabola.
To find the l-coordinate of the vertex, we can use the formula: l = -b/(2a), where a = -1 and b = 220.
l = -220/(2*(-1)) = 110
Substituting this value of l back into the equation for w from part (a), we can find w:
w = 220 - 110 = 110
Therefore, the dimensions that produce the greatest area are:
l = 110 meters
w = 110 meters
(a) To write w as a function of l, we can use the given information that the sum of l and w is 220 meters.
Since the sum of l and w is 220, we can write the equation as: l + w = 220.
To isolate w on one side, we can subtract l from both sides of the equation: w = 220 - l.
Therefore, w is a function of l and can be written as w = 220 - l.
(b) To write the area A as a function of l, we can use the formula for the area of a rectangle, which is length multiplied by width (A = l * w).
From part (a), we know that w = 220 - l. Substituting this value into the formula, we have:
A = l * (220 - l).
Therefore, the area A is a function of l and can be written as A = l * (220 - l).
(c) To find the dimensions that produce the greatest area, we need to find the value of l that maximizes the area function A = l * (220 - l).
To find the maximum value of the quadratic function A = l * (220 - l), we can take the derivative of A with respect to l and set it equal to zero:
dA/dl = 220 - 2l = 0.
Solving this equation, we have:
220 - 2l = 0
2l = 220
l = 110.
Therefore, the value of l that produces the greatest area is l = 110.
Substituting this value back into the equation w = 220 - l, we can find the value of w:
w = 220 - 110 = 110.
Therefore, the dimensions that produce the greatest area are:
l = 110
w = 110.
To answer these questions, let's break it down step by step.
(a) Write w as a function of l:
The sum of the length l and the width w is given as 220 meters. In other words, l + w = 220. To write w as a function of l, we need to isolate w on one side of the equation.
First, subtract l from both sides of the equation:
l + w - l = 220 - l
This simplifies to:
w = 220 - l
So, w is a function of l and can be expressed as w = 220 - l.
(b) Write the area A as a function of l:
The area of a rectangle is given by the formula A = length × width. We can substitute the function for w into this formula.
A = l × w
Substituting w = 220 - l, we have:
A = l × (220 - l)
Expanding the equation, we get:
A = 220l - l^2
So, the area A is a function of l and can be expressed as A = 220l - l^2.
(c) Find the dimensions that produce the greatest area:
To find the dimensions that produce the greatest area, we need to consider the quadratic equation A = 220l - l^2.
The equation is in the form of a downward-opening parabola, and the maximum point corresponds to the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where the equation is in the form Ax^2 + Bx + C.
In our case, the quadratic equation is -l^2 + 220l. Comparing it to the general form, we can see that a = -1, b = 220, and c = 0.
Using the formula x = -b / (2a), we have:
l = -220 / (2(-1))
Simplifying further:
l = 220 / 2
l = 110
So, the length l that produces the greatest area is 110 meters.
Now, using the function w = 220 - l, we can find the width:
w = 220 - 110
w = 110
Therefore, the dimensions that produce the greatest area are:
Length (l) = 110 meters
Width (w) = 110 meters