As fresh-poured concrete hardens, the chemical transformation releases energy at a rate of 2 W/kg. Assume that the center of a poured layer does not have any heat loss and that it has an average heat capacity of 0.9 kJ/kg K. Find the temperature rise during 1 h of the hardening (curing) process.

Use a CAS to carry out the following steps:
(a) Solve the equation $F'(x)=0$. Determine the intervals on which $F$ increases and the intervals on which $F$ decreases. Produce a figure that displays both the graph of $F$ and the graph of $F'$.
(b) Solve the equation $F''(x)=0$. Determine the intervals on which the graph of $F$ is concave up and the intervals on which the graph of $F$ is concave down. Produce a figure that displays both the graph of $F$ and the graph of $F''$. \

$$F(x)=\int_x^0 \sin 2t dt, \quad x \in[0,2 \pi]$$

If $8.40 \mathrm{~kJ}$ of heat is needed to raise the temperature of a sample of metal from $15^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C}$, how many kilojoules of heat will be required to raise the temperature of the same sample of metal from $25^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$ ?

Explain Le Chatelier's principle. Apply it to the following equilibria:
(a) $2 \mathrm{SO}_{2(\mathrm{~g})}+\mathrm{O} \rightleftharpoons 2 \mathrm{SO}_{3(g)}+$ Heat
(b) $\mathrm{N}_2 \mathrm{O}_{4(\mathrm{~g})} \rightleftharpoons 2 \mathrm{NO}_{2(\mathrm{~g})}-$ Heat

Predict the distance between an object and a concave mirror for each resulting image. Each distance should be a number relative to the mirror's focal length. -a bright beam of light _______

Air at $600 \mathrm{~K}$ flows with $3 \mathrm{~kg} / \mathrm{s}$ into a heat exchanger and out at $100^{\circ} \mathrm{C}$. How much $(\mathrm{kg} / \mathrm{s})$ water coming in at $100 \mathrm{kPa}, 20^{\circ} \mathrm{C}$ can the air heat to the boiling point?

The engine of a large ship does $2.00 \times 10^{8}\ \mathrm{J}$ of work with an efficiency of 5.00%. (a) How much heat transfer occurs to the environment? (b) How many barrels of fuel are consumed, if each barrel produces $6.00 \times 10^{9}\ \mathrm{J}$ of heat transfer when burned?

A student uses an immersion heater to heat 0.30 kg of water from $20^{\circ} \mathrm{C}$ to $80^{\circ} \mathrm{C}$ for tea. (a) If the heater is 75% efficient and takes 2.5 min to heat the water, what is its resistance? (b) How much current is in the heater? (Assume 120-V household voltage.)

A small, stationary sphere carries a net charge $Q$. You perform the following experiment to measure $Q$ : From a large distance you fire a small particle with mass $m=4.00 \times 10^{-4} \mathrm{~kg}$ and charge $q=5.00 \times 10^{-8} \mathrm{C}$ directly at the center of the sphere. The apparatus you are using measures the particle's speed $v$ as a function of the distance $x$ from the sphere. The sphere's mass is much greater than the mass of the projectile particle, so you assume that the sphere remains at rest. All of the measured values of $x$ are much larger than the radius of either object, so you treat both objects as point particles. You plot your data on a graph of $v^2$ versus $(1 / x)$. The straight line $v^2=400 \mathrm{~m}^2 / \mathrm{s}^2-$ $\left[\left(15.75 \mathrm{~m}^3 / \mathrm{s}^2\right) / x\right]$ gives a good fit to the data points.(a) Explain why the graph is a straight line.

In each part, sketch the graph of a function f with the stated properties.

f is increasing on $(−\infty,+\infty)$, has an inflection point at the origin, and is concave down on $(0,+\infty)$.

A rigid container is charged with butene, C4H8, and air in a stoichiometric ratio at P0, T0. The charge burns in a short time, with no heat transfer to state 2. The products then cool to 1200 K, state 3. Find the final pressure, P3, the total heat transfer, 1Q3, and the temperature immediately after combustion, T2.

You place an electric heater 3.00 m in front of a concave spherical mirror that has a focal length of 30.0 cm. Where would your hand feel warmest?

Consider the Gompertz differential equation

$$\frac{d P}{d t}=c P \ln \left(\frac{L}{P}\right)$$

where c is a positive constant and L is the carrying capacity of the environment.

Show that P(t) is increasing most rapidly when

$$t=\frac{\ln \ln \left(\frac{L}{P_0}\right)}{c}$$

Use determinants to solve each system of equations. Check your answers.

$$\begin{array}{l} 2 x-7 y=-18 \\ x-2 y=-6 \end{array}$$