lio A tennis ball has a radius of 3.5 cm . Three tennis balls fit exactly into a cylindrical tube. Using \( \pi=3 \cdot 14 \), calculate: (i) The volume of the three tennis balls (ii) The height of the tube (iii) The radius of the tube (iv) The volume of the tube, correct to two decimal places (v) The fraction of the volume of the tube taken up by the three tennis balls

(i) Each tennis ball is a sphere with radius \( r=3.5 \) cm. Its volume is \[ V_{\text{ball}} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (3.5)^3. \] Since \[ (3.5)^3 = \left(\frac{7}{2}\right)^3 = \frac{343}{8}, \] we have \[ V_{\text{ball}} = \frac{4}{3}\pi \cdot \frac{343}{8} = \frac{1372\pi}{24}. \] Taking \(\pi=3.14\), \[ V_{\text{ball}} \approx \frac{1372 \times 3.14}{24} \approx \frac{4308.08}{24} \approx 179.5\text{ cm}^3. \] Thus the volume of three tennis balls is \[ V_{\text{3 balls}} \approx 3\times 179.5 \approx 538.5\text{ cm}^3. \] (ii) The three tennis balls are stacked so that they just fit the cylindrical tube. Their diameters are \[ \text{Diameter} = 2r = 7 \text{ cm}, \] so the height of the tube is the sum of three diameters: \[ h = 3 \times 7 = 21 \text{ cm}. \] (iii) In order for each ball to touch the side of the tube, the inside radius of the tube must equal the radius of a tennis ball. Hence, \[ R = 3.5 \text{ cm}. \] (iv) The volume of the cylindrical tube is given by \[ V_{\text{tube}} = \pi R^2 h. \] Substitute \(R=3.5\) cm and \(h=21\) cm: \[ V_{\text{tube}} = 3.14 \times (3.5)^2 \times 21. \] We calculate \[ (3.5)^2 = 12.25, \] and then \[ 12.25 \times 21 = 257.25. \] Thus, \[ V_{\text{tube}} \approx 3.14 \times 257.25 \approx 807.77\text{ cm}^3. \] Rounded to two decimal places, the volume is approximately \(807.77\text{ cm}^3\). (v) The fraction of the tube’s volume taken up by three tennis balls is \[ \text{Fraction} = \frac{V_{\text{3 balls}}}{V_{\text{tube}}} \approx \frac{538.5}{807.77} \approx 0.667. \] This is equivalent to about \(\frac{2}{3}\). Thus, the answers are: (i) Volume of the three tennis balls: \(\approx 538.5\text{ cm}^3\). (ii) Height of the tube: \(21\text{ cm}\). (iii) Radius of the tube: \(3.5\text{ cm}\). (iv) Volume of the tube: \(\approx 807.77\text{ cm}^3\). (v) Fraction of the tube’s volume occupied by the tennis balls: approximately \(\frac{2}{3}\) (or 66.7%).