A jeweler resizes a ring so that its inner circumference is 8 centimeters. (a) What is the radius r (in cm) of the ring? r = cm (b) The inner circumference of the ring varies between 7.5 centimeters and 8.5 centimeters. How does the radius vary? < r < (c) Use the 𝜀–𝛿 definition of limit to describe this situation if 𝜀 = 0.5. (Round your value of 𝛿 to four decimal places.) lim r→ = 8; 𝜀 = 0.5; 𝛿 =
The Correct Answer and Explanation is:
Part (a): Finding the Radius of the Ring
To find the radius, we use the formula for the circumference of a circle:C=2πrC = 2 \pi rC=2πr
Where:
- CCC is the circumference,
- rrr is the radius,
- π≈3.1416\pi \approx 3.1416π≈3.1416.
The given inner circumference of the ring is 8 cm, so we can solve for rrr:8=2πr8 = 2 \pi r8=2πr
Divide both sides of the equation by 2π2\pi2π:r=82π=86.2832≈1.2732 cmr = \frac{8}{2\pi} = \frac{8}{6.2832} \approx 1.2732 \, \text{cm}r=2π8=6.28328≈1.2732cm
Thus, the radius rrr is approximately 1.2732 cm.
Part (b): Variation in the Radius
The problem states that the inner circumference varies between 7.5 cm and 8.5 cm. To determine how the radius varies, we apply the formula for the radius based on the circumference:r=C2πr = \frac{C}{2\pi}r=2πC
Let’s calculate the radius for the two extreme values of the circumference:
- For C=7.5 cmC = 7.5 \, \text{cm}C=7.5cm:
r=7.52π=7.56.2832≈1.1949 cmr = \frac{7.5}{2\pi} = \frac{7.5}{6.2832} \approx 1.1949 \, \text{cm}r=2π7.5=6.28327.5≈1.1949cm
- For C=8.5 cmC = 8.5 \, \text{cm}C=8.5cm:
r=8.52π=8.56.2832≈1.3526 cmr = \frac{8.5}{2\pi} = \frac{8.5}{6.2832} \approx 1.3526 \, \text{cm}r=2π8.5=6.28328.5≈1.3526cm
Thus, the radius varies between approximately 1.1949 cm and 1.3526 cm. This gives the range:1.1949<r<1.35261.1949 < r < 1.35261.1949<r<1.3526
Part (c): Using the ϵ\epsilonϵ-δ\deltaδ Definition of Limit
The ϵ\epsilonϵ-δ\deltaδ definition of the limit is a formal way of describing the behavior of a function as it approaches a particular value. Here, we are asked to use the definition to describe the situation where the radius approaches 1.2732 cm as the circumference approaches 8 cm.
The general form of the ϵ\epsilonϵ-δ\deltaδ definition of a limit is:For every ϵ>0, there exists a δ>0 such that if ∣C−8∣<δ, then ∣r−1.2732∣<ϵ.\text{For every } \epsilon > 0, \text{ there exists a } \delta > 0 \text{ such that if } |C – 8| < \delta, \text{ then } |r – 1.2732| < \epsilon.For every ϵ>0, there exists a δ>0 such that if ∣C−8∣<δ, then ∣r−1.2732∣<ϵ.
Given ϵ=0.5\epsilon = 0.5ϵ=0.5, we are tasked with finding the corresponding δ\deltaδ.
Start by writing the relationship between the radius rrr and the circumference CCC:r=C2πr = \frac{C}{2\pi}r=2πC
Now, we want to find δ\deltaδ such that:∣r−1.2732∣<0.5|r – 1.2732| < 0.5∣r−1.2732∣<0.5
Since r=C2πr = \frac{C}{2\pi}r=2πC, this becomes:∣C2π−1.2732∣<0.5\left|\frac{C}{2\pi} – 1.2732\right| < 0.52πC−1.2732<0.5
Simplifying:∣C−82π∣<0.5\left|\frac{C – 8}{2\pi}\right| < 0.52πC−8<0.5
Multiply both sides by 2π2\pi2π:∣C−8∣<0.5×2π≈3.1416|C – 8| < 0.5 \times 2\pi \approx 3.1416∣C−8∣<0.5×2π≈3.1416
Thus, the value of δ\deltaδ is approximately 3.1416. So, when ϵ=0.5\epsilon = 0.5ϵ=0.5, we can set δ=3.1416\delta = 3.1416δ=3.1416 to ensure that the radius rrr is within 0.5 cm of 1.2732 cm.
Summary of Answers:
- (a) The radius is approximately 1.2732 cm.
- (b) The radius varies between 1.1949 cm and 1.3526 cm.
- (c) Using the ϵ\epsilonϵ-δ\deltaδ definition, when ϵ=0.5\epsilon = 0.5ϵ=0.5, the corresponding δ\deltaδ is 3.1416.
