An aluminum pan whose thermal conductivity is 237 W/m^2$^\circ{}$C has a flat bottom whose diameter is 20 cm and thickness 0.6 cm. Heat is transferred steadily to boiling water in the pan through its bottom at a rate of 700 W. If the inner surface of the bottom of the pan is 105$^\circ{}$C, determine the temperature of the outer surface of the bottom of the pan.

The inner and outer surfaces of a 0.5-cm thick $2-m \times 2-m$ window glass in winter are $10^\circ C$ and $3^\circ C,$ respectively. If the thermal conductivity of the glass is $0.78 W/m \cdot K,$ determine the amount of heat loss through the glass over a period of 5 h. What would your answer be if the glass were 1 cm thick?

Find the exact arc length for the following problems over the given interval. Solve for the length of $x=y^{2} \text { from }(0,0) \text { to }(1,1).$ show that $x=(1 / 2) y^{2} \text { from }(0,0) \text { to }(2,2)$ is twice as long. Graph both functions and explain why this is so.

A 1.4-m-long, 0.2-cm-diameter electrical wire extends across a room that is maintained at 20 degree C. Heat is generated in the wire as a result of resistance heating, and the surface temperature of the wire is measured to be 240 degree C in steady operation. Also, the voltage drop and electric current through the wire are measured to be 110 V and 3 A, respectively. Disregarding any heat transfer by radiation, determine the convection heat transfer coefficient for heat transfer between the outer surface of the wire and the air in the room.

An aluminum saucepan with a diameter of $23 \mathrm{~cm}$ and a height of $6.0 \mathrm{~cm}$ is filled to the brim with water. The initial temperature of the pan and water is $19^{\circ} \mathrm{C}$. The pan is now placed on a stove burner and heated to $88^{\circ} \mathrm{C}$. Calculate the volume of water that overflows or the drop in water level in the pan, whichever is appropriate.

Water is boiled at 1 atm pressure in a 20-cm-internal diameter Teflon-pitted stainless-steel pan on an electric range. If it is observed that the water level in the pan drops by 10 cm in 15 min. determine the inner surface temperature of the pan.

An aluminum saucepan with a diameter of $23 \mathrm{~cm}$ and a height of $6.0 \mathrm{~cm}$ is filled to the brim with water. The initial temperature of the pan and water is $19^{\circ} \mathrm{C}$. The pan is now placed on a stove burner and heated to $88^{\circ} \mathrm{C}$. Will water overflow from the pan, or will the water level in the pan decrease? Explain.

A hot steam pipe having an inside surface temperature of $250^{\circ} \mathrm{C}$ has an inside diameter of $8 \mathrm{~cm}$ and a wall thickness of $5.5 \mathrm{~mm}$. It is covered with a $9-\mathrm{cm}$ layer of insulation having $k=0.5 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$, followed by a 4-cm layer of insulation having $k=0.25 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. The outside temperature of the insulation is $20^{\circ} \mathrm{C}$. Calculate the heat lost per meter of length. Assume $k=47 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$ for the pipe.

Let

$$\left\{ \mathbf { v } _ { 1 } , \dots , \mathbf { v } _ { n } \right\}$$

be a basis for a vector space V. Prove that

$$\left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } , \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } + \mathbf { v } _ { 3 } , \ldots , \mathbf { v } _ { 1 } + \cdots + \mathbf { v } _ { n } \right\}$$

is also a basis for V.

Find the flux of the vector field F across

$$\sigma$$

.

$$\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k } ; \sigma$$

is the portion of the cone

$$z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$$

between the planes z=1 and z=2, oriented by upward unit normals.

A 3-m-internal-diameter spherical tank made of 1 –cm-thick stainless steel is used to store iced water at $0^\circ C.$ The tank is located outdoors at $25^\circ C,$ Assuming the entire steel tank to be at $0^\circ C$ and thus the thermal resistance of the tank to be negligible, determine (a) the rate of heat transfer to the iced water in the tank and (b) the amount of ice at $0^\circ C$ that melts during a 24-hour period. The heat of fusion of water at atmospheric pressure is $h_{if}= 333.7 kJ/kg.$ The emissivity of the outer surface of the tank is 0.75, and the convection heat transfer coefficient on the outer surface can be taken to be $30W/m^2 \cdot K.$ Assume the average surrounding surface temperature for radiation exchange to be $15^\circ C.$

The inner and outer surfaces of a 0.5-cm-thick 2-m x 2-m window glass in winter are 15$^\circ{}$C and 6$^\circ{}$C, respectively. If the thermal conductivity of the glass is 0.78 W/m$^\circ{}$C, determine the amount of heat loss, in kJ, that occurs through the glass over a period of 10 h. What would your answer be if the glass were 1 cm thick?

The inner and outer surfaces of a 4-m × 7-m brick wall of thickness 30 cm and thermal conductivity $0.69 W/m \cdot K$ are maintained at temperatures of $26^\circ C$ and $8^\circ C,$ respectively. Determine the rate of heat transfer through the wall, in W.

A 60-gallon water heater is initially filled with water at $50^\circ F.$ Determine how much energy (in Btu) needs to be transferred to the water to raise its temperature to $120^\circ F.$ Evaluate the water properties at an average water temperature of $85^\circ F.$