A piece of wire x inches long is bent into the shape of an equilateral triangle. Find a function that represents the area of the triangle and state the domain of the function.
To find a function that represents the area of the equilateral triangle formed by a piece of wire x inches long, we can use the formula for the area of an equilateral triangle. The formula for the area (A) of an equilateral triangle with side length s is given by:
A = (√3/4) * s^2
In this case, since the wire is x inches long, the side length (s) of the equilateral triangle would be x/3 (since there are three equal sides in an equilateral triangle). So, we can substitute x/3 for s in the formula:
A = (√3/4) * (x/3)^2
= (√3/4) * (x^2/9)
= (√3/36) * (x^2)
Thus, the function that represents the area of the equilateral triangle formed by a piece of wire x inches long is:
A(x) = (√3/36) * (x^2)
The domain of this function would be the set of all non-negative real numbers, since the length of a wire cannot be negative and we are only considering real length measurements. So, the domain of the function A(x) is [0, ∞).
To find a function that represents the area of the equilateral triangle formed by a piece of wire x inches long, we can use the formula for the area of an equilateral triangle.
An equilateral triangle has all three sides of equal length, so each side of the triangle would be x/3 inches long.
The formula for the area of an equilateral triangle is given by:
Area = (sqrt(3) / 4) * side^2
Substituting x/3 for the side length, the function that represents the area, A(x), is:
A(x) = (sqrt(3) / 4) * (x/3)^2
Simplifying further, we have:
A(x) = (sqrt(3) / 4) * (x^2 / 9)
Therefore, the function representing the area of the equilateral triangle formed by the wire of length x is A(x) = (sqrt(3) / 4) * (x^2 / 9).
Now, let's consider the domain of the function. In this case, the domain represents the permissible values for x, which in turn represent the length of the wire.
The length of the wire cannot be negative because it represents a physical measurement. Hence, the domain of the function A(x) is all non-negative real numbers, i.e., x >= 0.
each side will be x/3 inches
sketch a height to create a 30-60-90° triangle
let the height be h
by ratios:
h/(x/3) = √3/2
h = (x/3)(√3/2) = (√3/6)x
area = (1/2)(base)(height)
= (1/2)(x/3)(√3/6)x
= (√3/36)x^2
check my arithmetic
or
h/(x/3) = sin60 = √3/2
h = ..... same as above