find the greatest possible area for the given perimeter for rectangular bumper car rides p=60in
Why did the bumper car install a circular perimeter fence?
Because it wanted to maximize the area of its "bumping" ground while minimizing any sharp corners!
To find the greatest possible area for a rectangular bumper car ride given a perimeter of 60 inches, we can use the formula for the perimeter of a rectangle, which is:
Perimeter = 2*(Length + Width)
Since we are given that the perimeter is 60 inches, we can write the equation as:
60 = 2*(Length + Width)
Now, we need to express one variable in terms of the other so we can find the greatest possible area. Let's solve this equation for Length:
Length + Width = 60/2
Length + Width = 30
Length = 30 - Width
Now, we can express the area of the rectangle in terms of Width:
Area = Length * Width
Substituting the value of Length we found:
Area = (30 - Width) * Width
To find the greatest possible area, we need to maximize this equation. Let's find the value of Width that will maximize the area by taking the derivative of the Area equation and setting it equal to zero:
d(Area)/d(Width) = 0
Differentiating the equation:
d(Area)/d(Width) = (30 - 2*Width)
Setting it equal to zero:
30 - 2*Width = 0
2*Width = 30
Width = 30/2
Width = 15 inches
Now, we have the width. We can substitute it back into the equation for Length:
Length = 30 - Width
Length = 30 - 15
Length = 15 inches
Therefore, the greatest possible area for a rectangular bumper car ride with a perimeter of 60 inches is achieved when the width is 15 inches and the length is also 15 inches.
Area = Length * Width
Area = 15 inches * 15 inches
Area = 225 square inches
To find the greatest possible area for a rectangular bumper car ride with a given perimeter of p = 60 inches, we can use the concept of optimization.
Let's assume the length of one side of the rectangular bumper car ride is x inches. Since the perimeter of a rectangle is given by the formula P = 2(l + w), where l represents the length and w represents the width, we can set up the equation:
60 = 2(x + w)
To find the area of a rectangle, we use the formula A = l * w. In this case, the area can be expressed as A = x * w.
Now, let's solve the equation for w in terms of x:
60 = 2(x + w)
30 = x + w
w = 30 - x
Substituting this value of w in the area formula:
A = x * (30 - x)
To find the greatest possible area, we need to maximize the function A. To do this, we can take the derivative of A with respect to x, set it equal to zero, and solve for x. However, since this is a quadratic function, we know that the maximum value occurs at the vertex of the parabola.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a represents the coefficient of the x^2 term and b represents the coefficient of the x term. In our case, the equation for A can be rewritten as:
A = -x^2 + 30x
Comparing it to the general form ax^2 + bx + c, we have a = -1 and b = 30. Plugging these values into the vertex formula:
x = -30 / (2 * (-1))
x = 15
Now that we have the value of x (one side of the rectangle), we can substitute it back into the equation for w:
w = 30 - x
w = 30 - 15
w = 15
So, the length of the rectangle is 15 inches and the width is also 15 inches. Therefore, the greatest possible area for the given perimeter of 60 inches is:
A = x * w
A = 15 * 15
A = 225 square inches
Thus, the greatest possible area for the given perimeter of the rectangular bumper car ride is 225 square inches.