Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. (Round your answers to the nearest hundredth.)
y = 7 - x^2
To find the dimensions of the rectangle of largest area, we need to analyze the problem and break it down into smaller steps. Here's how we can approach it:
Step 1: Understand the problem.
We are given a parabolic function y = 7 - x^2 and we need to find the rectangle with the largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola.
Step 2: Define the variables.
Let's define the variables that represent the dimensions of the rectangle. We need to find the length and width of the rectangle.
Let the length of the rectangle be L, and the width be W.
Step 3: Formulate the area of the rectangle.
The area of a rectangle is given by the formula A = Length x Width. In this case, we want to maximize the area.
Therefore, A = L * W.
Step 4: Express the width in terms of the length.
As specified in the problem, the base of the rectangle lies on the x-axis. So, the width of the rectangle will be given by the difference between the y-values of the parabolic curve at the two vertices of the rectangle. In other words, the width is determined by the parabola y = 7 - x^2.
So, the width W = (7 - x^2) - 0 = 7 - x^2.
Step 5: Express the area in terms of the length.
Now, we can substitute the expression for the width into the formula for area to get an expression for the area in terms of the length:
A = L * W = L * (7 - x^2)
Step 6: Find the derivative of the area function.
To find the maximum area, we need to find the critical point(s) of the area function. We can do this by finding the derivative of the area function with respect to the length L.
dA/dL = 7 - x^2
Step 7: Set the derivative equal to zero and solve for L.
To find the critical point(s), we set the derivative equal to zero and solve for L:
7 - x^2 = 0
Solving this equation, we get:
x = ±√7
Step 8: Determine the dimensions.
We have two critical points x = +√7 and x = -√7. We reject the negative value because we are looking for dimensions above the x-axis as specified in the problem statement.
So, we have x = √7.
Substituting this value into our expression for the width, we get:
W = 7 - (√7)^2 = 7 - 7 = 0
Since the width is 0, this means that the rectangle is degenerate and degenerates into a line segment.
And since the rectangle degenerates into a line segment, the dimensions are:
Length (L) = 2√7 (since the other side of the rectangle corresponds to the other x-value where the parabola intersects the x-axis)
Width (W) = 0
Therefore, the dimensions of the rectangle of largest area are approximately:
Length ≈ 5.29
Width ≈ 0
Note: It is important to round your answers to the nearest hundredth as stated in the question.
To find the dimensions of the rectangle of largest area, we need to determine the points on the parabola where the rectangle's vertices will be located.
First, let's find the x-coordinate of the vertex of the parabola. The vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b/2a.
In this case, the parabola is y = 7 - x^2, so a = -1 and b = 0. Plugging these values into the formula, we have x = 0.
Therefore, the x-coordinate of the vertex is 0.
Next, let's substitute this x-coordinate into the equation of the parabola to find the y-coordinate of the vertex.
Using x = 0, we have y = 7 - (0^2) = 7.
Therefore, the y-coordinate of the vertex is 7.
So, the vertex of the parabola is (0, 7).
Now, we can find the dimensions of the rectangle that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola.
The base of the rectangle will have a length of 2x, which means it runs from -x to x on the x-axis.
To find the x-values of the other two vertices, we need to solve the equation of the parabola for y = 0 (since they lie on the x-axis).
Setting y = 0 in the equation y = 7 - x^2, we have 0 = 7 - x^2.
Rearranging the equation, we get x^2 = 7.
Taking the square root of both sides, we have x = ±√7.
Therefore, the x-values of the other two vertices are x = -√7 and x = √7.
The length of the base is then 2x = 2√7.
To find the height of the rectangle, we need to find the difference between the y-coordinate of the vertex and the y-coordinate of one of the other vertices.
Using the y-coordinate of the vertex (7) and one of the other vertices, we have 7 - 0 = 7.
Therefore, the height of the rectangle is 7.
So, the dimensions of the rectangle of largest area are:
Base: 2√7
Height: 7
Note: The area of a rectangle is given by the formula A = base * height. Therefore, the area of the rectangle of largest area can be calculated by multiplying the base and height obtained above.
let P(x,y) be the point in quadrant I lying on the parabola.
Area = 2xy
= 2x(7-x^2)
= 14x - 2x^3
d(Area)/dx = 14 - 6x^2
= 0 for max of area
6x^2 = 14
...
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take it from there