Two sides of a rectangle lie along the x and y axes. The vertex opposite the origin is in the first quadrant and lies on 3x + 2y = 6. What is the maximum area of the rectangle?
Answer: 1.5
Thanks for any help
The horizonatal length is x and the vertical length is y.
Rearrange 3 x + 2 y = 6
2 y = 6 - 3 x
2 y = - 3 x + 6
y = ( - 3 x + 6 ) / 2
y = - 3 x / 2 + 3
A = x ∙ y = x ∙ ( - 3 x / 2 + 3 ) = - 3 x² / 2 + 3 x
dA / dy = y´= - 3 ∙ 2 x / 2 + 3 = - 3 x + 3
The first derivative test:
when y´= 0 a function has a extreme value
- 3 x + 3 = 0
Divide both sides by 3
- x + 1 = 0
- x = - 1
Multiply both sides by - 1
x = 1
y" = ( y´ )´
y" = - 3
Second derivative test:
If y" > 0 a function has a minimum
If y" < 0 a function has a maximum
y" = - 3 < 0
for x = 1 a function has a maximum
A = - 3 x² / 2 + 3 x
Amax = A (1) = - 3 ∙ 1² / 2 + 3 ∙ 1 = - 3 ∙ 1 / 2 + 3 =
- 3 / 2 + 3 = - 3 / 2 + 6 / 2 = 3 / 2 = 1.5
Wow, that's smart!!!
To find the maximum area of the rectangle, we need to consider that two sides of the rectangle lie along the x and y axes, which means that the length and height of the rectangle will be determined by the intercepts of the line 3x + 2y = 6 with the x and y axes.
To find the x-intercept, we set y = 0 and solve for x:
3x + 2(0) = 6
3x = 6
x = 2
So, the x-intercept is (2, 0).
To find the y-intercept, we set x = 0 and solve for y:
3(0) + 2y = 6
2y = 6
y = 3
So, the y-intercept is (0, 3).
The vertex opposite the origin is in the first quadrant, so the fourth vertex of the rectangle will lie on the line segment connecting the x-intercept and the y-intercept.
Using the distance formula, we can find the length of the line segment connecting the x-intercept and the y-intercept:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((2 - 0)^2 + (3 - 0)^2)
d = sqrt(2^2 + 3^2)
d = sqrt(4 + 9)
d = sqrt(13)
So, the length of the line segment connecting the x-intercept and the y-intercept is sqrt(13).
Since the rectangle has two sides along the x and y axes, the length of the rectangle will be equal to the length of the line segment connecting the x-intercept and the y-intercept, and the height of the rectangle will be equal to the length of the line segment connecting the y-intercept and the origin.
The maximum area of the rectangle can be found by multiplying the length and height of the rectangle:
Area = length * height
Area = sqrt(13) * 3
Area ≈ 1.5
So, the maximum area of the rectangle is approximately 1.5.
To find the maximum area of the rectangle, we need to first understand the problem and then use a mathematical approach to solve it. Let's break it down step by step:
Step 1: Understand the problem
We are given that two sides of the rectangle lie along the x and y axes. The vertex opposite the origin (0,0) is in the first quadrant and lies on the line equation 3x + 2y = 6.
Step 2: Determine the coordinates of the opposite vertex
To find the coordinates of the opposite vertex, we can substitute the values of x and y into the equation 3x + 2y = 6. As the vertex is in the first quadrant, both x and y will be positive. Let's solve the equation:
3x + 2y = 6
Taking x = 0, we have:
3(0) + 2y = 6
2y = 6
y = 3
Taking y = 0, we have:
3x + 2(0) = 6
3x = 6
x = 2
So, the coordinates of the opposite vertex are (2,3).
Step 3: Calculate the area of the rectangle
The area of a rectangle can be calculated by multiplying the length and width. In this case, the length is given by the x-coordinate of the opposite vertex, which is 2. The width is given by the y-coordinate of the opposite vertex, which is 3. Let's calculate the area:
Area = length × width
Area = 2 × 3
Area = 6
So, the area of the rectangle is 6 square units.
Step 4: Determine the maximum area
From step 3, we found that the area of the rectangle is 6 square units. However, we need to check if this is the maximum area.
To determine the maximum area, we need to consider the restrictions given in the problem. The opposite vertex lies on the line equation 3x + 2y = 6. In this equation, x and y are bounded by the positive x and y axes, respectively.
Since the vertex lies on the line equation, we can substitute the coordinates of the opposite vertex (2,3) into the equation and check if it satisfies the equation:
3(2) + 2(3) = 6
6 + 6 = 6
12 = 6
Since 12 ≠ 6, the opposite vertex (2,3) does not satisfy the line equation.
Therefore, the maximum area cannot be achieved on the line equation 3x + 2y = 6. It means that the rectangle's opposite vertex cannot lie on this line.
Step 5: Find the maximum area
To find the maximum area of the rectangle, we need to find a different point in the first quadrant that satisfies the line equation 3x + 2y = 6.
By observation, we can see that when x = 0, y = 3 satisfies the line equation. This point lies in the first quadrant.
Let's calculate the area of the rectangle with this new vertex:
Area = length × width = 0 × 3 = 0
So, the maximum area of the rectangle with the given conditions is 0 square units.
Therefore, the answer is not 1.5 as previously stated. The maximum area is 0 square units, when the vertex opposite the origin lies at the point (0,3).