A rectangular framed picture has a total length and width of 20 cm
20 cm and 10 cm, respectively. The frame has width x cm.
a Find the rule for the area (A cm ) of the picture inside.
b What are the minimum and maximum values of x?
A = (20-2x)(10-2x) = 4(10-x)(5-x)
The roots are at x=5,10
But, for x > 5, 5-x < 0 which is not allowed
so, assuming an actual width of both the frame and the picture,
So, 0 < x < 5
a) To find the area of the picture inside the frame, we need to subtract the area of the frame from the total area of the frame and picture combined.
Step 1: Find the total area of the frame and picture combined.
The length of the frame and picture combined is 20 cm, and the width is 10 cm. Therefore, the total area is:
Total Area = Length * Width = 20 cm * 10 cm = 200 cm²
Step 2: Find the area of the frame.
The frame has a width of x cm. Since it is a rectangular frame, the area of the frame can be calculated as:
Area of Frame = Length of Frame * Width of Frame = (20 cm + 2x) * (10 cm + 2x) = 200 cm² + 40x + 4x²
Step 3: Subtract the area of the frame from the total area to get the area of the picture inside.
Area of Picture Inside = Total Area - Area of Frame = 200 cm² - (200 cm² + 40x + 4x²) = -40x - 4x²
Therefore, the rule for the area of the picture inside is A = -4x² - 40x.
b) To find the minimum and maximum values of x, you need to consider the constraints of the problem. The width of the frame cannot be negative, so x must be greater than or equal to zero.
To find the minimum value of x, we need to find when the area of the picture inside, A, is at its lowest. The area of a rectangle is always non-negative, so the minimum value of x would be when the area, -4x² - 40x, is equal to zero. Solving this quadratic equation will give us the minimum value of x.
To find the maximum value of x, we consider the width of the frame, which cannot exceed the total width of the rectangular frame and picture combined. In this case, the maximum value of x is 5 cm since any value larger than that would exceed the width of the entire rectangular frame and picture.
So, the minimum value of x is 0 cm, and the maximum value of x is 5 cm.