A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m, find the rectangle's dimensions that will produce the largest ar... A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m, find the rectangle's dimensions that will produce the largest area of the window.
Understand the Problem
The question is asking for the dimensions of a rectangle that, when combined with an equilateral triangle above it, will produce the largest possible area for the given perimeter of 12 meters.
Answer
The dimensions of the rectangle are $x = \frac{6}{3 - \frac{\sqrt{3}}{2}}$ meters for the base and $h = \frac{12 - 3x}{2}$ meters for the height.
Answer for screen readers
The dimensions of the rectangle that will produce the largest area of the window are:
- Base: $x = \frac{6}{3 - \frac{\sqrt{3}}{2}}$ meters
- Height: $h = \frac{12 - 3x}{2}$ meters
Steps to Solve
- Define Variables
Let the base of the rectangle be $x$ meters and the height of the rectangle be $h$ meters. The triangle above the rectangle will have a base of $x$ meters.
- Set Up the Perimeter Equation
The perimeter of the window consists of the rectangle and the equilateral triangle. The perimeter equation can be set up as: $$ P = 2h + 2x + x = 12 $$ Simplifying this gives us: $$ 2h + 3x = 12 $$
- Express Height in Terms of Base
From the perimeter equation, solve for $h$: $$ 2h = 12 - 3x $$ $$ h = \frac{12 - 3x}{2} $$
- Area of the Window
The total area $A$ of the window, which includes the rectangle and the triangle, can be expressed as: $$ A = \text{Area of rectangle} + \text{Area of triangle} $$ $$ A = (x)(h) + \frac{1}{2}(x)(\text{height of the triangle}) $$ Since the height of the equilateral triangle is $\frac{\sqrt{3}}{2}x$, we have: $$ A = x \cdot h + \frac{1}{2}x \cdot \frac{\sqrt{3}}{2}x $$
- Substitute for Height
Now substitute $h$ from the previous result: $$ A = x \cdot \frac{12 - 3x}{2} + \frac{\sqrt{3}}{4}x^2 $$ $$ A = \frac{12x - 3x^2}{2} + \frac{\sqrt{3}}{4}x^2 $$ Combine terms: $$ A = 6x - \frac{3}{2}x^2 + \frac{\sqrt{3}}{4}x^2 $$ $$ A = 6x - \left(\frac{3}{2} - \frac{\sqrt{3}}{4}\right)x^2 $$
- Find the Critical Points
To maximize the area, take the derivative of $A$ with respect to $x$: $$ \frac{dA}{dx} = 6 - \left(3 - \frac{\sqrt{3}}{2}\right)x $$ Set the derivative to zero to find critical points: $$ 6 - \left(3 - \frac{\sqrt{3}}{2}\right)x = 0 $$ Solve for $x$: $$ x = \frac{6}{3 - \frac{\sqrt{3}}{2}} $$
- Calculate Dimensions
Calculate the value of $h$ using the value of $x$ obtained.
- Final Values for Dimensions
Substituting the value of $x$ into $h = \frac{12 - 3x}{2}$ to find the height $h$.
The dimensions of the rectangle that will produce the largest area of the window are:
- Base: $x = \frac{6}{3 - \frac{\sqrt{3}}{2}}$ meters
- Height: $h = \frac{12 - 3x}{2}$ meters
More Information
This problem combines optimization with geometry. Maximizing the area while adhering to a fixed perimeter constraint helps understand how dimensions impact area. The equilateral triangle above the rectangle introduces additional geometric properties.
Tips
- Forgetting to consider the total perimeter involving both the rectangle and the triangle.
- Not correctly expressing the area in terms of a single variable.
- Making algebraic errors when simplifying expressions.
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