A rectangle is inscribed in an equilateral triangle so that one side of the rectangle lies on the base of the triangle. How do I find the maximum area of the rectangle when the triangle has side length of 10?
First, let's look at a picture.
Some initial observations:

The area
#A# of the rectangle is#A=bh# . 
By symmetry, the base of the triangle is of length
#b+2t# , and thus, as it is of length#10# , we have#b+2t = 10 => t = 5b/2# 
If we decide
#b# that also determines#h# , and thus we can write#h# as a function of#b# .
To write
Solving for
Then, we can rewrite our formula for the area as
If we look at the graph for
And so the vertex, and thus the maximum area, is at
Finally, we calculate the area from this to get
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To find the maximum area of the rectangle inscribed in an equilateral triangle with a side length of 10, we can use geometry and optimization techniques.
Let's denote the side length of the equilateral triangle as ( 10 ) units. Since the rectangle is inscribed in the triangle, its height will be the same as the height of the equilateral triangle, which is ( \frac{\sqrt{3}}{2} ) times the side length. Therefore, the height of the rectangle is ( 5\sqrt{3} ) units.
Now, let's denote the width of the rectangle as ( x ) units. The rectangle's width will vary along the base of the equilateral triangle.
The area ( A ) of the rectangle can be expressed as the product of its width and height:
[ A = x \times 5\sqrt{3} ]
Since the rectangle is inscribed in the triangle, the sum of its width and the remaining length on the base of the triangle will be equal to the side length of the equilateral triangle:
[ x + x + x = 10 ]
Solving for ( x ), we get:
[ x = \frac{10}{3} ]
Substitute this value of ( x ) into the expression for the area ( A ):
[ A = \frac{10}{3} \times 5\sqrt{3} ]
[ A = \frac{50\sqrt{3}}{3} ]
Therefore, the maximum area of the rectangle inscribed in the equilateral triangle with a side length of ( 10 ) units is ( \frac{50\sqrt{3}}{3} ) square units.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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