A rubber band is stretched around the four circles (each with radius r cm).
i) Show that the total length of the rubber band is given by the expression r(8+2π)cm.
ii) Show that the area of the shape is given by the expression r^2(12+π) cm^2
Consider two of the circles. The rubber band stretches between two parallel radii, separated by a distance of 2r.
In addition, it stretches 1/4 of the war around a circle. That distance is π/2 r.
Since there are 4 circles, the total length is 4(2r + π/2 r) = r(8+2π)
Use similar reasoning to get the area of the square minus the 4 corners.
Thanks :>
To solve this problem, we need to break it down into smaller steps. Let's start with part (i) where we need to show that the total length of the rubber band is given by the expression r(8+2π) cm.
i) To find the total length of the rubber band, we first need to find the circumference of each circle. Recall that the circumference of a circle with radius r is given by the formula 2πr.
Since there are four circles, the total length of the rubber band will be the sum of the circumferences of each circle. Let's calculate that:
Circumference of the first circle = 2πr cm
Circumference of the second circle = 2πr cm
Circumference of the third circle = 2πr cm
Circumference of the fourth circle = 2πr cm
Total length of the rubber band = Sum of the circumferences of each circle
Total length of the rubber band = (2πr + 2πr + 2πr + 2πr) cm
Total length of the rubber band = 8πr cm
However, we need to express the answer in terms of r(8+2π). To do this, we can factor out r from the equation:
Total length of the rubber band = r(8π) cm + r(2π) cm
Total length of the rubber band = r(8 + 2π) cm
Therefore, the total length of the rubber band is given by the expression r(8 + 2π) cm. This completes part (i) of the problem.
ii) Now let's move on to part (ii) where we need to show that the area of the shape is given by the expression r^2(12 + π) cm^2.
To find the area of the shape, we need to calculate the individual areas of the four circles and then subtract the overlapping areas.
Area of the first circle = πr^2 cm^2
Area of the second circle = πr^2 cm^2
Area of the third circle = πr^2 cm^2
Area of the fourth circle = πr^2 cm^2
Total area of the shape = Sum of the areas of each circle - Overlapping areas
Total area of the shape = (πr^2 + πr^2 + πr^2 + πr^2) cm^2
Total area of the shape = 4πr^2 cm^2
Again, we need to express the answer in terms of r^2(12 + π). To do this, we can factor out r^2 from the equation:
Total area of the shape = r^2(4π) cm^2
Total area of the shape = r^2(12 + π) cm^2
Therefore, the area of the shape is given by the expression r^2(12 + π) cm^2. This completes part (ii) of the problem.