an isolation triangle has a base of 6 units and a height of 12 units. find the maximum possible area for a rectangle that is inscribed in the triangle and has one side resting on the base of the triangle. what are the dimensions of the rectangle of maximum area?
72
To find the maximum possible area for a rectangle inscribed in an isolation triangle, we need to determine the dimensions of the rectangle.
Step 1: Draw the isolation triangle and label the base and height.
The base of the triangle is given as 6 units, and the height is given as 12 units.
Step 2: Label the dimensions of the rectangle.
Let the width of the rectangle be 'x' units (resting on the base), and the height of the rectangle be 'y' units.
Step 3: Use similar triangles to relate the dimensions of the rectangle to the dimensions of the triangle.
Since the rectangle is inscribed in the triangle, the larger triangle formed by connecting the top vertices of the rectangle to the top vertex of the triangle is similar to the smaller triangle formed by connecting the bottom vertices of the rectangle to the base of the triangle.
The bases of these two triangles are in the ratio of x:6, and the heights are in the ratio of y:12.
So, we can write the equation:
x/6 = y/12
Step 4: Solve the equation for y.
Multiply both sides of the equation by 12:
12(x/6) = y
Simplify:
2x = y
Step 5: Calculate the area of the rectangle.
The area of a rectangle is given by the formula A = length × width.
In this case, the length is the height of the rectangle, which is y units, and the width is the base of the rectangle, which is x units.
So, the area A can be written as:
A = x × y
Substituting the value of y from step 4:
A = x × 2x
Simplifying:
A = 2x^2
Step 6: Find the maximum area.
To find the maximum area, we need to find the value of x that maximizes the expression 2x^2.
Step 7: Take the derivative of the expression with respect to x and set it equal to zero to find the critical points.
dA/dx = 4x
4x = 0
x = 0
Step 8: Determine whether the critical point is a maximum or minimum.
Since the isolation triangle has a finite area, x cannot be equal to zero. This means that there is no critical point.
Step 9: Consider the boundary values.
Since x must be greater than 0 and less than or equal to 6 (the length of the base of the triangle), we have 0 < x ≤ 6.
Step 10: Evaluate the area function at the boundary values.
Let's calculate the area at the boundary values to determine the maximum area.
When x = 0, the area is 0.
When x = 6, the area is:
A = 2(6)^2
A = 2(36) = 72
So, the maximum possible area for a rectangle inscribed in the isolation triangle is 72 square units.
Step 11: Find the dimensions of the rectangle that maximizes the area.
We can substitute the value of x that corresponds to the maximum area (6) into our equation from step 4 to find the corresponding value of y:
y = 2x = 2(6) = 12
Therefore, the dimensions of the rectangle with the maximum area are 6 units by 12 units.
To find the maximum possible area for a rectangle inscribed in the isolation triangle, we need to understand that maximizing the area of the rectangle means making it as large as possible.
Step 1: Visualize the problem
First, let's visualize the isolation triangle with its base of 6 units and height of 12 units. Draw a rectangle inside the triangle, making one side rest on the base.
_________
| : |
|---:-----|
12 | : |
^ | : |
| | : |
|__|___:_____|
6 Base
The rectangle has two sides parallel to the triangle's base and two sides perpendicular to it.
Step 2: Understand the properties of the rectangle
Let the two sides of the rectangle parallel to the base have lengths x units and y units, respectively.
Step 3: Determine the area of the rectangle
The area A of a rectangle is calculated by multiplying its length (base) by its width (height). In this case, it will be xy.
Step 4: Express the width of the rectangle in terms of x
Since one side of the rectangle rests on the base of the triangle, its width will be equal to x.
Step 5: Determine the length of the rectangle in terms of x
To express the length of the rectangle in terms of x, we need to find the equation of the hypotenuse of the isolation triangle. The equation is given by the Pythagorean theorem: a² + b² = c².
In this case, the base (b) is 6 units, and the height (a) is 12 units. The hypotenuse (c) represents the length of the rectangle. Plugging the values into the equation, we get:
6² + 12² = c²
36 + 144 = c²
180 = c²
√180 = c
c = √(36 * 5)
c = √36 * √5
c = 6√5
Therefore, the length of the rectangle is 6√5 units.
Step 6: Express the area of the rectangle in terms of x only
Now that we know the length of the rectangle is 6√5, we can express the area A in terms of x only.
A = x * (6√5)
A = 6x√5
Step 7: Find the maximum area
To find the maximum possible area, we need to determine the value of x that maximizes A.
Step 8: Differentiate A with respect to x
To find the optimal value of x, we differentiate A with respect to x and set it equal to 0.
dA/dx = 6√5
Step 9: Set dA/dx = 0 and solve for x
6√5 = 0
Since this equation has no solution, we conclude that there is no maximum possible area for the rectangle inside the isolation triangle.
Therefore, there are no dimensions for a rectangle of maximum area.
In conclusion, a rectangle inscribed in an isolation triangle with a base of 6 units and a height of 12 units does not have a maximum possible area.
X=6(1-y/12)
X=6-0.5y
A=xy
dA/dx=d/dx(6-0.5y)(y)
dA/dx=6-y
0=6-y
y=6
X=6-(0.5)(6)
X=3