A pair of eyeglass frames are made of an epoxy plastic (coefficient of linear expansion = 1.30 10-4°C−1). At room temperature (20.0°C), the frames have circular lens holes 2.27 cm in radius. To what temperature must the frames be heated if lenses 2.28 cm in radius are to be inserted into them?

L1= 2• π• r =2• π • 2.27•10^-2 = 0.1426 m

L2 = 2• π• R =2• π • 2.28•10^-2 = 0.1433 m
ΔL = 6.566•10^-4 m
α = (ΔL/L)/ ΔT,
ΔT=(ΔL/L)/ α = (6.566•10^-4/0.1426)/1.3•10^-4 =35.42oC
T = T(o) + ΔT =20 +35.42 = 55.42 oC

Well, you know what they say, the frames need to warm up to embrace the bigger lenses! But let's get to the calculations. We need to find the temperature at which the frames will expand enough to accommodate the larger lenses.

The change in radius (ΔR) can be calculated using the formula:

ΔR = α * R * ΔT

Where α is the coefficient of linear expansion, R is the original radius (2.27 cm), and ΔT is the change in temperature.

In this case, we need ΔR to be 0.01 cm (2.28 cm - 2.27 cm). Let's substitute the given values:

0.01 cm = (1.30 * 10^(-4) °C^(-1)) * (2.27 cm) * ΔT

Simplifying that equation, we can find ΔT:

ΔT ≈ (0.01 cm) / ((1.30 * 10^(-4) °C^(-1)) * (2.27 cm))

So, ΔT is approximately equal to 1.17 * 10^2 °C.

Therefore, the frames need to be heated to approximately 117 °C to accommodate the larger lenses. Just remember to handle the hot frames with care and don't get too heated yourself!

To find the temperature to which the frames must be heated if lenses of a different radius are to be inserted, we can use the concept of linear expansion.

The formula for linear expansion is given by:
ΔL = α * L * ΔT

Where:
ΔL is the change in length
α is the coefficient of linear expansion
L is the initial length
ΔT is the change in temperature

In this case, we are given the coefficient of linear expansion (α) and the initial radius of the lens hole (L), and we need to find the change in temperature (ΔT) to accommodate lenses of a different radius.

1. Convert the radius from centimeters to meters:
Initial radius (L) = 2.27 cm = 0.0227 m

2. Calculate the change in radius:
ΔL = New radius (2.28 cm) - Initial radius (0.0227 m)
ΔL = 0.0228 m - 0.0227 m = 0.0001 m

3. Rearrange the linear expansion formula to solve for ΔT:
ΔT = ΔL / (α * L)

4. Substitute the given values:
ΔT = 0.0001 m / (1.30 * 10^(-4) °C^(-1) * 0.0227 m)

5. Calculate the change in temperature:
ΔT ≈ 38.325 °C

6. Add the change in temperature to the room temperature (20.0°C) to find the final temperature:
Final temperature = Room temperature + ΔT
Final temperature ≈ 20.0 °C + 38.325 °C

Therefore, the frames must be heated to approximately 58.325 °C to accommodate lenses with a radius of 2.28 cm.

To find the temperature to which the frames must be heated for the lenses to fit, we need to consider the coefficient of linear expansion of the epoxy plastic and the change in radius of the lens holes.

Here's how you can solve the problem step by step:

Step 1: Determine the change in radius of the lens holes.
The original radius of the lens holes is given as 2.27 cm, and we need to calculate the change in radius when it becomes 2.28 cm. The change in radius can be calculated as follows:
Change in radius = final radius - initial radius = 2.28 cm - 2.27 cm = 0.01 cm.

Step 2: Calculate the change in temperature of the frames.
The change in temperature can be calculated using the formula:
Change in temperature = change in length × coefficient of linear expansion × initial temperature.

In this case, since the radius is changing, we need to calculate the change in length. The change in length is given by:
Change in length = 2 × π × initial radius × change in radius.

Substituting the given values:
Change in length = 2 × π × 2.27 cm × 0.01 cm ≈ 0.142 cm.

Now, using the coefficient of linear expansion (1.30 × 10^(-4) °C^(-1)) and the initial temperature (20.0 °C), we can calculate the change in temperature:
Change in temperature = 0.142 cm × 1.30 × 10^(-4) °C^(-1) × 20.0 °C = 0.000368 °C.

Step 3: Find the final temperature.
To find the final temperature to which the frames must be heated, we need to add the change in temperature to the initial temperature:
Final temperature = initial temperature + change in temperature = 20.0 °C + 0.000368 °C.

Therefore, to insert lenses with a radius of 2.28 cm into the frames, the frames must be heated to approximately 20.00037 °C.

Note: When rounding off the temperature, make sure to keep enough significant figures to maintain accuracy.