Grade 12 · Washer volumes · 42 of 600
Washer volumes · Outer radius factor a=3; Inner radius factor b=0.4
Use 0≤x≤1.2 and inspect the cross-section at x=1.2. Calculate the outer radius, washer area, and accumulated volume. Givens: Outer radius factor a=3; Inner radius factor b=0.4.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Rotate the region between R(x)=3x and r(x)=0.4x around the x-axis. Both are radii, with R≥r≥0. Use 0≤x≤1.2 and inspect the cross-section at x=1.2. Calculate the outer radius, washer area, and accumulated volume.
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A starting point
The washer area is π(R²−r²), not π(R−r)². Integrate this area in x.
Work through the reasoning
Step 1
Identify the model and target
Rotate the region between R(x)=3x and r(x)=0.4x around the x-axis. Both are radii, with R≥r≥0. Use 0≤x≤1.2 and inspect the cross-section at x=1.2. The governing relation is R=ax, r=bx; V(T)=π(a²−b²)T³/3. The washer area is π(R²−r²), not π(R−r)². Integrate this area in x.
Step 2
Substitute and calculate
At x=1.2, π(R²−r²)=π[(3)²−(0.4)²](1.2)²=39.991218. Integrating from 0 to 1.2 gives π[3²−0.4²](1.2)³/3=15.996487.
Step 3
Check the mathematical meaning
The inner radius is 0.48 and outer radius is 3.6. The area is nonnegative. Since the area is proportional to x², volume equals endpoint area times 1.2/3=15.996487. Upper x bound: 1.2; Outer radius: 3.6; Cross-section area: 39.991; Accumulated volume: 15.996. Decimal values are rounded, so use unrounded intermediate values.
The answer
At x=1.2, π(R²−r²)=π[(3)²−(0.4)²](1.2)²=39.991218. Integrating from 0 to 1.2 gives π[3²−0.4²](1.2)³/3=15.996487. The inner radius is 0.48 and outer radius is 3.6. The area is nonnegative. Since the area is proportional to x², volume equals endpoint area times 1.2/3=15.996487. Animation check: Upper x bound: 1.2; Outer radius: 3.6; Cross-section area: 39.991; Accumulated volume: 15.996. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
Rotating a region between two radius curves around an axis produces annular cross-sections. Each washer's area is the outer disk area minus the inner disk area. Squaring the difference of radii would incorrectly remove the cross term and describe a different area. Rotate the region between R(x)=3x and r(x)=0.4x around the x-axis. Both are radii, with R≥r≥0. Use 0≤x≤1.2 and inspect the cross-section at x=1.2. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
R=ax, r=bx; V(T)=π(a²−b²)T³/3
03 · Reflect and transfer
Explain what changes and why.
If both radius factors double while the x interval stays fixed, by what factor does the volume change?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.