48. A target board for a dart game has been designed as three concentric circles where each coloured region is the same area. If the radius of the blue circle is \( r \mathrm{~cm} \) and the radius of the outer circle is 10 cm , determine the value of \( r \).

To find the value of \( r \), we need to ensure that the three regions corresponding to the concentric circles have equal areas. The area of a circle is given by the formula \( A = \pi r^2 \). 1. The area of the outer circle with a radius of 10 cm is: \[ A_{\text{outer}} = \pi (10^2) = 100\pi \, \text{cm}^2 \] 2. We can denote the area of the blue circle as: \[ A_{\text{blue}} = \pi r^2 \] 3. The area of the ring between the blue circle and the outer circle (the red region) is: \[ A_{\text{red}} = A_{\text{outer}} - A_{\text{blue}} = 100\pi - \pi r^2 = (100 - r^2)\pi \] 4. Since all three areas (the inner blue circle, the red ring, and the outer ring) need to be equal, we can set up the following equations: \[ A_{\text{blue}} = A_{\text{red}} = A_{\text{outer ring}} \] 5. Now, we are aware that the area of each region is equal. Therefore, we can express the equal area condition as \( \pi r^2 = (100 - r^2)\pi = A_{\text{outer ring}} \). 6. To find \( r \), we can express the total area of the three equal regions as \( 3A = 100\pi \), yielding \( A = \frac{100\pi}{3} \). 7. Now, this translates to \( \pi r^2 = \frac{100\pi}{3} \) which simplifies to: \[ r^2 = \frac{100}{3} \] \[ r = \sqrt{\frac{100}{3}} \approx 5.77\, \text{cm} \] Thus, \( r \) is approximately \( 5.77 \, \text{cm} \).