once ryan used his volume calculations to pack his drums away in the van, he headed off to the airport.

he quickly realised he had forgotten to calculate the weight of his drums, which needs to be less than $25\text{ kg}$ for the airport to accept it on board the plane.

calculate the weight of ryans drums (ignoring cymbals and stands), and decide whether or not he can afford to transport his drum kit.

the dimensions of the drums are given below in the format $\text{diameter} \times \text{length}$:

\
\begin{aligned}
v_{\text{snare}} &= (36\text{ cm} \times 10\text{ cm}) \\
v_{\text{high tom}} &= (25\text{ cm} \times 18\text{ cm}) \\
v_{\text{mid tom}} &= (30\text{ cm} \times 20\text{ cm}) \\
v_{\text{floor tom}} &= (41\text{ cm} \times 50\text{ cm}) \\
v_{\text{kick drum}} &= (56\text{ cm} \times 46\text{ cm})
\end{aligned}
\

we will approximate the weight of the drum kit by ignoring the metal hoops and lugs, and just calculate the weight of the wooden drum shells. the radius of each drum extends right out to the edge of the drum shell. each drum has the same shell thickness, which is given below, and are made from the wood maple, which has a particular density (weight per volume) also given below.

shell thickness: $2\text{ cm}$
maple density: $700\text{ kg/m}^3$

give your answer in appropriate units and round to 2 decimal places.

Question

once ryan used his volume calculations to pack his drums away in the van, he headed off to the airport.

he quickly realised he had forgotten to calculate the weight of his drums, which needs to be less than $25\text{ kg}$ for the airport to accept it on board the plane.

calculate the weight of ryans drums (ignoring cymbals and stands), and decide whether or not he can afford to transport his drum kit.

the dimensions of the drums are given below in the format $\text{diameter} \times \text{length}$:

\
\begin{aligned}
v_{\text{snare}} &= (36\text{ cm} \times 10\text{ cm}) \\
v_{\text{high tom}} &= (25\text{ cm} \times 18\text{ cm}) \\
v_{\text{mid tom}} &= (30\text{ cm} \times 20\text{ cm}) \\
v_{\text{floor tom}} &= (41\text{ cm} \times 50\text{ cm}) \\
v_{\text{kick drum}} &= (56\text{ cm} \times 46\text{ cm})
\end{aligned}
\

we will approximate the weight of the drum kit by ignoring the metal hoops and lugs, and just calculate the weight of the wooden drum shells. the radius of each drum extends right out to the edge of the drum shell. each drum has the same shell thickness, which is given below, and are made from the wood maple, which has a particular density (weight per volume) also given below.

shell thickness: $2\text{ cm}$
maple density: $700\text{ kg/m}^3$

give your answer in appropriate units and round to 2 decimal places.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Hollow Cylinder Volume", "Unit Conversion" ], "new_concepts": [ "Density Mass Calculation" ], "current_concepts": [ "Hollow Cylinder Volume", "Unit Conversion", "Density Mass Calculation" ] } ## Calculate total volume of wooden shells Using the *Hollow Cylinder Volume* knowledge point $$ \begin{aligned} t &= 2\text{ cm} = 0.02\text{ m}\ V_{\text{shell}} &= \pi (R^2 - r^2) L = \pi (R^2 - (R - t)^2) L = \pi (2 R t - t^2) L\ V_{\text{snare}} &= \pi (2 \times 0.18 \times 0.02 - 0.02^2) \times 0.10 = 0.00068\pi\text{ m}^3\ V_{\text{high tom}} &= \pi (2 \times 0.125 \times 0.02 - 0.02^2) \times 0.18 = 0.000828\pi\text{ m}^3\ V_{\text{mid tom}} &= \pi (2 \times 0.15 \times 0.02 - 0.02^2) \times 0.20 = 0.00112\pi\text{ m}^3\ V_{\text{floor tom}} &= \pi (2 \times 0.205 \times 0.02 - 0.02^2) \times 0.50 = 0.0039\pi\text{ m}^3\ V_{\text{kick drum}} &= \pi (2 \times 0.28 \times 0.02 - 0.02^2) \times 0.46 = 0.005152\pi\text{ m}^3\ V_{\text{total}} &= (0.00068 + 0.000828 + 0.00112 + 0.0039 + 0.005152)\pi = 0.01168\pi\text{ m}^3 \approx 0.0366938\text{ m}^3 \end{aligned} $$ ## Calculate total weight of the drums Using the *Unit Conversion* knowledge point $$ \begin{aligned} ho &= 700\text{ kg/m}^3\ W &= V_{\text{total}} \times ho\ W &= 0.0366938\text{ m}^3 \times 700\text{ kg/m}^3 \approx 25.6857\text{ kg} \end{aligned} $$ ## Determine transport eligibility Compare calculated weight to limit. $$ \begin{aligned} W &\approx 25.69\text{ kg}\ 25.69\text{ kg} &> 25\text{ kg} \end{aligned} $$ The total weight of Ryan's drums is **$25.69\text{ kg}$**. Since this weight is greater than the limit of $25\text{ kg}$, he **cannot** afford to transport his drum kit on board the plane. { "subject": "Mathematics", "question_type": "Word Problem", "knowledge_point": [ "Mathematics", "Geometry", "Hollow Cylinder Volume" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate total volume of wooden shells

Calculate total weight of the drums

Determine transport eligibility

Answer: