Can't figure this out... :(
An isosceles trapezium with its longest side facing upward has an area of 145cm^2. Find the amount of material needed to make up the sides of the trapezium, when the side facing upward does not need to be covered (i.e. Perimeter - length of longest side).
I assume you'd sub 145 as the area into the trapezium area rule a= h((a+b)/2)
Or perhaps divide it into a rectangle and 2 triangles?? Any help appreciated!
I see no way to do this. All you know is the area. There are many isosceles trapezoids that can have such an area, each with a different perimeter.
For instance, if the height is 5, then (a+b)/2 = 29
and the slant sides are of length
s^2 = 5^2 + ((b-a)/2)^2
So, the perimeter can have many possible values. And that's just using h=5.
Something more needs to be nailed down.
To solve this problem, we can use a combination of the trapezium area formula and the perimeter formula for an isosceles trapezium.
Let's start by using the trapezium area formula, which is:
Area = (1/2) * (sum of the parallel sides) * height
In this case, we know the area is 145cm^2, and the height is undetermined (let's call it h). The sum of the parallel sides can be calculated by subtracting the length of the longest side (let's call it L) from the perimeter of the trapezium.
Now, let's denote the lengths of the equal sides as a and b. Since it is an isosceles trapezium, we have a = b.
Since the longest side is facing upward, let's call the length of the bottom side c.
The perimeter of the trapezium can be calculated as:
Perimeter = a + b + L + c
Now, we need to substitute these values into the formula for the area:
145 = (1/2) * (a + b) * h
We also know that the perimeter can be expressed as:
Perimeter = 2a + L + c
Given that the side facing upward does not need to be covered, the amount of material needed is equal to the perimeter minus the length of the longest side:
Material needed = Perimeter - L
Material needed = (2a + L + c) - L
Material needed = 2a + c
To find the amount of material needed, we need to eliminate the extra variables. Let's start by expressing a and b in terms of c:
a = b (equal sides of an isosceles trapezium)
Perimeter = 2a + L + c
Perimeter = 2b + L + c
Now, let's express b in terms of a:
Perimeter = 2a + L + c
Perimeter = 2a + L + c
We can rewrite the equation for the area using a and c:
145 = (1/2) * (a + a) * h
145 = a * h
Now, we have two equations:
145 = a * h
Perimeter = 2a + L + c
To solve for a and c, we need a value for h. Unfortunately, the problem does not provide the value for h. Without this information, it is not possible to determine the amount of material needed to make up the sides of the trapezium.