Problem:
Consider the surfaces defined by:
z = 8 − x² − y² and z = x² + y².
These two surfaces intersect along a circle in the xy-plane.
Question:
If the constant term in the first equation increases by 100% (i.e., changes from 8 to 16), by what percentage does the radius of the circle of intersection increase?
Answer Choices:
- 41.4%
- 50%
- 100%
- 82.8%
Answer:
A) 41.4%
Explanation:
When the constant term in the first equation increases by 100%, it changes from 8 to 16. Here’s how this affects the radius of the circle of intersection:
-
Original Intersection:
- Given Equations:
- z = 8 − x² − y²
- z = x² + y²
- Finding the Intersection:
8 − x² − y² = x² + y²
8 = 2x² + 2y²
x² + y² = 4 - Radius of Intersection Circle (r₁):
r₁ = √4 = 2
- Given Equations:
-
After Increasing the Constant Term by 100%:
- New First Equation:
z = 16 − x² − y²
- Finding the New Intersection:
16 − x² − y² = x² + y²
16 = 2x² + 2y²
x² + y² = 8 - Radius of New Intersection Circle (r₂):
r₂ = √8 = 2√2 ≈ 2.828
- New First Equation:
-
Calculating the Percentage Increase in Radius:
Percentage Increase = ((r₂ − r₁) / r₁) × 100%
= ((2.828 − 2) / 2) × 100%
= (0.828 / 2) × 100%
= 0.414 × 100% = 41.4%
Summary:
When the constant term in the equation z = 8 − x² − y² increases by 100% to become 16, the radius of the circle where the two surfaces intersect increases from 2 to approximately 2.828. This results in a 41.4% increase in the radius of the intersection circle.