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Anya has two different sized cylindrical coffee mugs. The larger mug has an internal height of 15 centimeters (cm), and the smaller mug has an internal height of 10cm. Both mugs have an internal diameter of 8cm. Which of the following is closest to the difference in cubic centimeters (cm^(3)) between the internal volume of the larger mug and the internal volume of the smaller mug? Choose 1 answer: (A) 63cm^(3) (B) 251cm^(3) (c) 754cm^(3) (D) 1,005cm^(3)

Anya has two different sized cylindrical coffee mugs. The larger mug has an internal height of 1515 centimeters (cm) (\mathrm{cm}) , and the smaller mug has an internal height of 10 cm 10 \mathrm{~cm} . Both mugs have an internal diameter of 8 cm 8 \mathrm{~cm} . Which of the following is closest to the difference in cubic centimeters (cm3) \left(\mathrm{cm}^{3}\right) between the internal volume of the larger mug and the internal volume of the smaller mug?\newlineChoose 11 answer:\newline(A) 63 cm3 63 \mathrm{~cm}^{3} \newline(B) 251 cm3 251 \mathrm{~cm}^{3} \newline(C) 754 cm3 754 \mathrm{~cm}^{3} \newline(D) 1,005 cm3 1,005 \mathrm{~cm}^{3}

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Q. Anya has two different sized cylindrical coffee mugs. The larger mug has an internal height of 1515 centimeters (cm) (\mathrm{cm}) , and the smaller mug has an internal height of 10 cm 10 \mathrm{~cm} . Both mugs have an internal diameter of 8 cm 8 \mathrm{~cm} . Which of the following is closest to the difference in cubic centimeters (cm3) \left(\mathrm{cm}^{3}\right) between the internal volume of the larger mug and the internal volume of the smaller mug?\newlineChoose 11 answer:\newline(A) 63 cm3 63 \mathrm{~cm}^{3} \newline(B) 251 cm3 251 \mathrm{~cm}^{3} \newline(C) 754 cm3 754 \mathrm{~cm}^{3} \newline(D) 1,005 cm3 1,005 \mathrm{~cm}^{3}
  1. Calculate Volume: First, we need to calculate the volume of each mug. The formula for the volume of a cylinder is V=πr2hV = \pi r^2 h, where rr is the radius and hh is the height. The radius is half the diameter, so for both mugs, the radius is 8 cm/2=4 cm8 \, \text{cm} / 2 = 4 \, \text{cm}.
  2. Volume of Larger Mug: Now, let's calculate the volume of the larger mug using the formula. The height of the larger mug is 15 cm15\,\text{cm}. Volume of larger mug = π×(4 cm)2×15 cm\pi \times (4\,\text{cm})^2 \times 15\,\text{cm}
  3. Volume of Smaller Mug: Perform the calculation for the larger mug:\newlineVolume of larger mug = π×16 cm2×15 cm\pi \times 16 \text{ cm}^2 \times 15 \text{ cm}\newlineVolume of larger mug = π×240 cm3\pi \times 240 \text{ cm}^3
  4. Find Difference: Next, calculate the volume of the smaller mug. The height of the smaller mug is 10 cm10\,\text{cm}. Volume of smaller mug = π×(4 cm)2×10 cm\pi \times (4\,\text{cm})^2 \times 10\,\text{cm}
  5. Simplify Difference: Perform the calculation for the smaller mug:\newlineVolume of smaller mug = π×16 cm2×10 cm\pi \times 16 \, \text{cm}^2 \times 10 \, \text{cm}\newlineVolume of smaller mug = π×160 cm3\pi \times 160 \, \text{cm}^3
  6. Calculate Numerical Value: Now, we find the difference in volume between the two mugs:\newlineDifference in volume = Volume of larger mug - Volume of smaller mug\newlineDifference in volume = π×240 cm3−π×160 cm3\pi \times 240 \, \text{cm}^3 - \pi \times 160 \, \text{cm}^3
  7. Compare with Options: Simplify the difference in volume:\newlineDifference in volume = π×(240 cm3−160 cm3)\pi \times (240 \, \text{cm}^3 - 160 \, \text{cm}^3)\newlineDifference in volume = π×80 cm3\pi \times 80 \, \text{cm}^3
  8. Compare with Options: Simplify the difference in volume:\newlineDifference in volume = π×(240 cm3−160 cm3)\pi \times (240 \, \text{cm}^3 - 160 \, \text{cm}^3)\newlineDifference in volume = π×80 cm3\pi \times 80 \, \text{cm}^3Since π\pi is approximately 3.141593.14159, we can now calculate the numerical value of the difference in volume:\newlineDifference in volume ≈3.14159×80 cm3\approx 3.14159 \times 80 \, \text{cm}^3\newlineDifference in volume ≈251.3272 cm3\approx 251.3272 \, \text{cm}^3
  9. Compare with Options: Simplify the difference in volume:\newlineDifference in volume = π×(240 cm3−160 cm3)\pi \times (240 \, \text{cm}^3 - 160 \, \text{cm}^3)\newlineDifference in volume = π×80 cm3\pi \times 80 \, \text{cm}^3Since π\pi is approximately 3.141593.14159, we can now calculate the numerical value of the difference in volume:\newlineDifference in volume ≈3.14159×80 cm3\approx 3.14159 \times 80 \, \text{cm}^3\newlineDifference in volume ≈251.3272 cm3\approx 251.3272 \, \text{cm}^3Looking at the options provided, the closest value to our calculated difference in volume is 251 cm3251 \, \text{cm}^3, which corresponds to option (B).

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