Shellie pays $4.00 for a square piece of wood, which she makes into a stop sign by cutting the corners off. what is the cost of the wasted part?
I don't even know where to begin?
do I take $4.00*1/8....which equals half and subtract that from 4.00...getting $2.50
Check this out:
http://en.wikipedia.org/wiki/Stop_sign
It looks like 2/9 of the original square gets removed.
Each of the cut-off corners would be an isosceles right-angled triangle.
The stop sign will be an octagon with each side equal to x.
Consider one of the triangles, the hypotenuse will be x.
by Pythagoras you can show that each of the equal sides must be x/√2
each side of the original square is then equal to
x/√2 + x + x/√2 = x(2+√2)/2
and the original area = x^2(2+√2)^2/4 = x^2(3+2√2)/2
area of one triangle = (1/2)(x/√2)(x/√2) = x^2/4
or the total wasted part is x^2
so the part wasted is x^2/(x^2(3+2√2)/2))($4)
= 8/(3+2√2) dollars or appr. $1.37
typo!
last part should be .....
each side of the original square is then equal to
x/√2 + x + x/√2 = x(2+√2)/√2
and the original area = x^2(2+√2)^2/2 = x^2(3+2√2)
area of one triangle = (1/2)(x/√2)(x/√2) = x^2/4
or the total wasted part is x^2
so the part wasted is x^2/(x^2(3+2√2))($4)
= 4/(3+2√2) dollars or appr. $0.69
Reiny is correct for a regular octagon. In looking at the figure of
http://en.wikipedia.org/wiki/Stop_sign
I erroneously assumed that each pair of cut-off corners amounted to 1/9 of the square. That would amoount to $0.89 wasted.
To calculate the cost of the wasted part, we first need to determine the shape of the stop sign and then calculate the area of the wasted part.
Since Shellie starts with a square piece of wood and cuts off the corners to make a stop sign, we need to visualize the shape of the resulting stop sign. A stop sign typically has eight sides, including an octagonal shape.
To find the cost of the wasted part, we need to calculate the area of the removed corners. In this case, it involves finding the area of four triangles (since four corners were cut off) and subtracting that from the original square.
Here's how you can calculate it step-by-step:
1. Determine the length of each side of the square piece of wood. Let's assume it is "s" units.
2. Calculate the area of the original square. The formula for the area of a square is A = s^2. In this case, the area is s * s = s^2.
3. Calculate the length of one of the cut-off corners. Since the corners were cut off evenly, each corner has the same length. Let's call this length "c."
4. Calculate the area of one triangle. The formula for the area of a triangle is A = (base * height) / 2. In this case, the base of the triangle is equal to the length of the cut-off corner, which is "c." The height of the triangle is equal to the length of one side of the original square, which is "s." Thus, the area of one triangle is A = (c * s) / 2.
5. Calculate the area of all four triangles. Since four corners were cut off, we need to multiply the area of one triangle by four. Therefore, the area of all four triangles is A = 4 * [(c * s) / 2] = 2 * c * s.
6. Calculate the area of the wasted part. To find the area of the wasted part, subtract the area of all four triangles from the original square. The formula is Wasted Area = Area of Square - Area of All Four Triangles = s^2 - 2 * c * s.
7. Calculate the cost of the wasted part. Assuming the cost of the wood is evenly distributed, we can calculate the cost of the wasted part by multiplying the wasted area by the cost per unit area. In this case, the cost is $4.00 for the original square, so the cost per unit area is $4.00 / s^2 (since the area of the square is s^2).
To summarize, the formula to calculate the cost of the wasted part is:
Cost of Wasted Part = (s^2 - 2 * c * s) * ($4.00 / s^2)
Remember to substitute the values of "s" and "c" to get the precise answer.
Note: Without specific values for side length and corner length, we cannot provide an exact answer.