4. Determine how much the oil level in a cylindrical tank will rise with an increase in temperature from 15^circ C to 40^circ C The density of oil at 15^circ C is 900kg/m^3 Tank diameter d=10m oil fills the reservoir at 15^circ C to a height of H=12 m. Thermal expansion coefficient of oil beta _(T)=6,4cdot 10^-41/degree expansion is not taken into account. V_(15)=pi R^2H rho _(t)=(rho _(15))/(1+beta _(T)(t-15))arrow Delta V=beta _(T)VDelta Tarrow Delta V=pi (d^2)/(4)Delta H

To determine how much the oil level in the cylindrical tank will rise with an increase in temperature from $15^{\circ }C$ to $40^{\circ }C$, we can use the formula for thermal expansion:<br /><br />$\Delta V = \beta_{T} V \Delta T$<br /><br />where $\Delta V$ is the change in volume, $\beta_{T}$ is the thermal expansion coefficient, $V$ is the initial volume, and $\Delta T$ is the change in temperature.<br /><br />Given that the diameter of the tank is $d = 10m$ and the oil fills the reservoir to a height of $H = 12m$ at $15^{\circ }C$, we can calculate the initial volume of the oil:<br /><br />$V_{15} = \pi R^{2} H$<br /><br />where $R$ is the radius of the tank, which is half of the diameter:<br /><br />$R = \frac{d}{2} = \frac{10}{2} = 5m$<br /><br />Substituting the values, we get:<br /><br />$V_{15} = \pi (5)^{2} \cdot 12 = 300\pi m^{3}$<br /><br />Now, we can calculate the change in temperature:<br /><br />$\Delta T = 40^{\circ }C - 15^{\circ }C = 25^{\circ }C$<br /><br />Substituting the values into the thermal expansion formula, we get:<br /><br />$\Delta V = \beta_{T} V \Delta T = 6.4 \times 10^{-4} \cdot 300\pi \cdot 25 = 15.1\pi m^{3}$<br /><br />Finally, we can calculate the change in height by dividing the change in volume by the cross-sectional area of the tank:<br /><br />$\Delta H = \frac{\Delta V}{\pi R^{2}} = \frac{15.1\pi}{\pi (5)^{2}} = 1.21m$<br /><br />Therefore, the oil level in the cylindrical tank will rise by approximately 1.21 meters with an increase in temperature from $15^{\circ }C$ to $40^{\circ }C$.