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 3 cm and an area equal to one sixth of the area of the rectangle. Find the required height of the diamond.
Why did the car manufacturer go with a diamond logo? Because they wanted to show that their cars are a "cut above" the rest!
Now, let's tackle this problem.
We know that the diamond's horizontal width is 3 cm. To find the required height of the diamond, we need to determine the area of the rectangle, find one-sixth of that area, and then use that to calculate the height.
Since the diamond is inside the rectangle, its area will be less than or equal to the area of the rectangle. Let's call the height of the rectangle 'h.'
The area of the rectangle is given by the formula A = length × width. In this case, the length of the rectangle is unknown, so we'll just leave it as 'L'.
The area of the diamond is one-sixth of the area of the rectangle. Therefore, the area of the diamond (AD) is one-sixth of the area of the rectangle (AR):
AD = (1/6) × AR
Let's denote the height of the diamond as 'd' and its width as 'w'. We have been given that the width of the diamond is 3 cm, so w = 3 cm.
The area of the diamond is equal to its width multiplied by its height: AD = w × d.
Now we can set up an equation:
w × d = (1/6) × AR
Substituting in the values we know:
3 cm × d = (1/6) × AR
Simplifying:
d = (1/6) × AR ÷ 3 cm
The height of the rectangle is 'h', and the width is 3 cm. Therefore, the area of the rectangle is AR = h × 3 cm.
Substituting this value back into the equation:
d = (1/6) × (h × 3 cm) ÷ 3 cm
The centimeters cancel out:
d = (1/6) × h
The required height of the diamond is (1/6) times the height of the rectangle.
So, the required height of the diamond is a sixth of the height of the rectangle, but I hope it doesn't feel down in the dumps being smaller than the rectangle. Remember, diamonds are beautiful in all shapes and sizes!
To find the required height of the diamond, we need to determine the dimensions of the rectangle first.
Let's assume the width of the rectangle is "w" cm, and its height is "h" cm.
The area of the rectangle is equal to the product of its width and height: Area = w * h.
Given that the diamond has a horizontal width of 3 cm, its vertical height will be represented by "x" cm.
The area of the diamond is equal to one-sixth of the area of the rectangle: Area of diamond = (1/6) * Area of rectangle.
Since the diamond is in the shape of a rectangle, the area of the diamond can be calculated as the product of its width and height: Area of diamond = 3 cm * x cm.
Now, we can equate the two areas and solve the equation:
(1/6) * Area of rectangle = 3 cm * x cm
Substituting the equation for the area of the rectangle, we have:
(1/6) * (w * h) = 3 cm * x cm
Simplifying the equation:
w * h = 18 cm * x
Since the diamond is in the shape of a rectangle, the width of the rectangle is equal to the width of the diamond: w = 3 cm.
Substituting this value into the equation:
3 cm * h = 18 cm * x
Dividing both sides by 3 cm:
h = 6 cm * x
Therefore, the required height of the diamond is 6 cm times the height of the rectangle.
To find the required height of the diamond, we first need to determine the area of the rectangle and the area of the diamond.
Let's assume the height of the rectangle is h cm. Since we know the diamond is inside the rectangle, we can say that the length of the rectangle is greater than or equal to the width of the diamond. Therefore, the length of the rectangle is equal to or greater than 3 cm.
The area of a rectangle is calculated by multiplying its length by its width. So, the area of the rectangle is given by:
Area of rectangle = Length × Width
Since the width of the diamond is 3 cm, we can say that the area of the rectangle is:
Area of rectangle = Length × 3
Now, the area of the diamond is one sixth (1/6) of the area of the rectangle. Therefore, we can write this relation as:
Area of diamond = (1/6) × Area of rectangle
We can substitute the area of the rectangle from the earlier equation into this relation:
Area of diamond = (1/6) × (Length × 3)
Since the diamond is in the shape of a square, its area is given by:
Area of diamond = Width × Height
Since the width of the diamond is given as 3 cm, we can write this as:
Area of diamond = 3 × Height
Now, we can equate the two equations for the area of the diamond:
3 × Height = (1/6) × (Length × 3)
Simplifying this equation, we get:
3 × Height = (1/6) × 3 × Length
3 × Height = (1/6) × 3 × h
3 × Height = (1/2) × h
Now, we can isolate the height by dividing both sides of the equation by 3:
Height = (1/2) × h ÷ 3
Height = h/6
Therefore, the required height of the diamond is equal to one-sixth of the height of the rectangle.
If the diamond is just a rotated square, then its diagonal is 3, so its area is 9/2.
So, the rectangle has width 3 and area 27, so its height is 9.