find the area of the largest rectangle having one side on the x axis and inscribed in the triangle formed by the lines y=x, y=0, and 3x + y = 20
if the height of the rectangle is y, the base of the rectangle is (20-y)/3 - y.
So, the area is y((20-y)/3 - y) = 4/3 (5y-y^2)
max area where 5-2y = 0, or y = 5/2.
So, the largest rectangle has area 50/3
To find the area of the largest rectangle inscribed in the given triangle, you can follow these steps:
1. Start by finding the vertices of the triangle formed by the lines y = x, y = 0, and 3x + y = 20.
- The intersection point of y = x and y = 0 is (0,0).
- To find the intersection point of y = x and 3x + y = 20, substitute y = x into the second equation and solve for x:
3x + x = 20
4x = 20
x = 5
Thus, the intersection point is (5,5).
2. Determine the length of the base of the triangle, which is the horizontal side of the rectangle.
- The base of the triangle is the difference between the x-coordinates of the two vertices:
Base = 5 - 0 = 5
3. Calculate the height of the triangle by finding the y-coordinate of the vertex on the line 3x + y = 20.
- Substitute x = 5 into 3x + y = 20 and solve for y:
3(5) + y = 20
15 + y = 20
y = 5
Therefore, the height of the triangle is 5.
4. Determine the area of the triangle.
- The area of a triangle is given by the formula: Area = (base * height) / 2
Area = (5 * 5) / 2
Area = 25 / 2
Area = 12.5
5. Since the largest rectangle is inscribed in the triangle, its base length is equal to the base length of the triangle, which is 5.
6. To find the height of the rectangle, consider that the diagonal of the rectangle is a segment joining two vertices of the triangle.
- The two vertices on the triangle are (0,0) and (5,5).
- Using the distance formula, calculate the distance between these two points to find the diagonal of the rectangle, which is also the height of the rectangle.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((5 - 0)^2 + (5 - 0)^2)
= sqrt(5^2 + 5^2)
= sqrt(50)
= 5 * sqrt(2)
7. Since the rectangle is inscribed in the triangle, its diagonal (height) should be perpendicular to the base of the triangle. This means the rectangle is a square, so its base and height are equal.
- Hence, the height of the rectangle is also 5 * sqrt(2).
8. Finally, calculate the area of the rectangle which is equal to the product of its base and height.
Area of rectangle = base * height
= 5 * (5 * sqrt(2))
= 25 * sqrt(2)
≈ 35.36
Therefore, the area of the largest rectangle inscribed in the given triangle is approximately 35.36 square units.