Pregunta
upstudy study bank question image url

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

Ask by Macdonald Hart. in Ireland
Mar 09,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

(i) Volume of three tennis balls: ≈538.5 cm³ (ii) Height of the tube: 21 cm (iii) Radius of the tube: 3.5 cm (iv) Volume of the tube: ≈807.77 cm³ (v) Fraction of the tube’s volume occupied by the tennis balls: ≈66.7%

Solución

(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%).

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

The Deep Dive

To find the volume of three tennis balls, we start with the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \] Given the radius \( r = 3.5 \) cm and \( \pi = 3.14 \): (i) The volume of one tennis ball is: \[ V = \frac{4}{3} \cdot 3.14 \cdot (3.5)^3 = \frac{4}{3} \cdot 3.14 \cdot 42.875 \approx 179.59 \text{ cm}^3 \] Thus, the volume of three tennis balls is approximately: \[ 3 \cdot 179.59 \approx 538.77 \text{ cm}^3 \] (ii) Since three tennis balls fit exactly in the tube, the height of the tube is three times the diameter of one tennis ball. The diameter \( d = 2 \cdot 3.5 = 7 \) cm, so the height is: \[ 3 \cdot 7 \text{ cm} = 21 \text{ cm} \] (iii) The radius of the tube is equal to the radius of one tennis ball, which is: \[ 3.5 \text{ cm} \] (iv) The volume of the cylindrical tube is given by the formula: \[ V_{\text{tube}} = \pi r^2 h \] Substituting the values in: \[ \pi = 3.14, r = 3.5 \text{ cm}, h = 21 \text{ cm} \] \[ V_{\text{tube}} = 3.14 \cdot (3.5)^2 \cdot 21 = 3.14 \cdot 12.25 \cdot 21 \approx 831.47 \text{ cm}^3 \] (v) The fraction of the volume of the tube taken up by the three tennis balls is: \[ \text{Fraction} = \frac{\text{Volume of balls}}{\text{Volume of tube}} = \frac{538.77}{831.47} \approx 0.65 \] In conclusion: (i) Volume of three tennis balls: \(\approx 538.77 \text{ cm}^3\) (ii) Height of the tube: \(21 \text{ cm}\) (iii) Radius of the tube: \(3.5 \text{ cm}\) (iv) Volume of the tube: \(\approx 831.47 \text{ cm}^3\) (v) Fraction of volume occupied: \(\approx 0.65\) or \(65\%\).

preguntas relacionadas

Latest Geometry Questions

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad