91Ó°ÊÓ

First, we will find the initial height of the coffee in the cup by dividing its volume by the area of a circle with the given diameter. Initial Height of Coffee (h) = Initial Volume of Coffee (V) / Base area of the cup. The area of the cup's base can be calculated using the formula: A = πr² The radius of the cup (r) is half of its diameter, so r = 7.00 cm / 2 = 3.50 cm. Base area of the cup (A) = π(3.50 cm)^2 \( = 38.4845 cm^2\) Initial Height of Coffee (h) = \( \frac{350 cm^3}{38.4845 cm^2} = 9.096 cm\)

We are now going to get some numbers to make a reasonable assumption, we will consider the coefficient of linear expansion for water (\( \alpha \)) of \(10^{-4}/°C\). When the temperature changes, the height and the radius of the cup for the coffee will change too. We can calculate the change in height and radius due to thermal contraction using the formula: ΔL = L × α × ΔT L represents the initial dimension (either height or the radius), ΔL is the change in dimension, α is the coefficient of linear expansion, and ΔT is the temperature change. Temperature change (ΔT) = Final temperature - Initial temperature = 45.0 °C - 95.0 °C = -50.0 °C Change in height (Δh) = h × α × ΔT = 9.096 cm × (1 × \(10^{-4}/°C\)) × (-50.0 °C) = -0.04548 cm Change in radius (Δr) = r × α × ΔT = 3.50 cm × (1 × \(10^{-4}/°C\)) × (-50.0 °C) = -0.01750 cm Now we have the changes in height and radius, we can calculate the new height and radius of the filled part of the cup when the temperature drops to 45.0 °C: Final Height of filled part (h') = Initial Height of Coffee (h) + Change in height (Δh) = 9.096 cm + (-0.04548 cm) = 9.05052 cm Final Radius of the filled part (r') = Initial Radius of the cup (r) + Change in radius (Δr) = 3.50 cm + (-0.01750 cm) = 3.48250 cm Now that we have the final height and radius, we can calculate the final volume of the filled part: Final Volume of the filled part (V') = Base area of the filled part × Final Height of the filled part Final Base Area = π × (r')^2 = π × (3.48250 cm)^2 = \(37.773 cm^2\) Final Volume of the filled part (V') = \( 37.773 cm^2 × 9.05052 cm = 342.098 cm^3\)

To calculate the decrease in level due to thermal contraction, we need to find the difference between the initial height and the final height: Decrease in coffee level = Initial Height of Coffee - Final Height of the filled part = 9.096 cm - 9.05052 cm = 0.04548 cm Since we want our answer in millimeters, we will convert the decrease in coffee level to millimeters: Decrease in coffee level = 0.04548 cm × 10 = 0.4548 mm Now, we can compare this result to the given observation of a 3.00 mm drop in the coffee level. The calculated decrease due to thermal contraction (0.4548 mm) is much less than the observed decrease (3.00 mm). Therefore, it's clear that the primary cause of the decrease in coffee level is not due to thermal contraction.