Grade 12 · Washer volumes · 306 of 600
Washer volumes · Outer radius factor a=1.5; Inner radius factor b=0.8
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=1.5; Inner radius factor b=0.8.
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)=1.5x and r(x)=0.8x 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)=1.5x and r(x)=0.8x 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[(1.5)²−(0.8)²](1.2)²=7.283468. Integrating from 0 to 1.2 gives π[1.5²−0.8²](1.2)³/3=2.913387.
Step 3
Check the mathematical meaning
The inner radius is 0.96 and outer radius is 1.8. The area is nonnegative. Since the area is proportional to x², volume equals endpoint area times 1.2/3=2.913387. Upper x bound: 1.2; Outer radius: 1.8; Cross-section area: 7.283; Accumulated volume: 2.913. Decimal values are rounded, so use unrounded intermediate values.
The answer
At x=1.2, π(R²−r²)=π[(1.5)²−(0.8)²](1.2)²=7.283468. Integrating from 0 to 1.2 gives π[1.5²−0.8²](1.2)³/3=2.913387. The inner radius is 0.96 and outer radius is 1.8. The area is nonnegative. Since the area is proportional to x², volume equals endpoint area times 1.2/3=2.913387. Animation check: Upper x bound: 1.2; Outer radius: 1.8; Cross-section area: 7.283; Accumulated volume: 2.913. 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.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
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)=1.5x and r(x)=0.8x 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.