a new car manufacturer is designing a logo. it is in the shape of a diamond inside a rectangle. the diamond is to have a horizontal width of 3cm and an area equal to one-sixth of the area of the rectangle. find the required height of the diamond
To find the required height of the diamond, we first need to find the area of the rectangle and then calculate one-sixth of that area.
Let's assume the width of the rectangle is w and the height of the rectangle is h. Since the diamond is inside the rectangle, we know that the width of the diamond is equal to the width of the rectangle (3cm in this case). Let's call the height of the diamond "x".
The area of the rectangle is calculated by multiplying its width by its height, so Area of Rectangle = w * h.
We are given that the area of the diamond is one-sixth of the area of the rectangle, so the Area of Diamond = (1/6) * (w * h).
However, we also know that the width of the diamond is equal to the width of the rectangle, so the width of the diamond is also equal to w.
The area of a diamond is calculated by multiplying its diagonal lengths and dividing by 2, so taking the width of the diamond (w) and the height of the diamond (x), the Area of Diamond = (w * x) / 2.
Equating the two expressions for the Area of the Diamond, we have:
(w * h) / 6 = (w * x) / 2
To find the height of the diamond (x), we can solve this equation. First, let's cross multiply:
2 * (w * h) = 6 * (w * x)
Now, divide both sides by 6w:
(2 * h) / 6 = x
Simplifying this expression further, we have:
h / 3 = x
Therefore, the required height of the diamond is equal to one-third of the height of the rectangle.