A detective finds a murder victim at 9 am. The temperature of the body is measured at $90.3^{\circ} \mathrm{F}$. One hour later, the temperature of the body is $89.0^{\circ} \mathrm{F}$. The temperature of the room has been maintained at a constant $68^{\circ} \mathrm{F}$. Solve the differential equation to estimate the time the murder occurred.

Choose a name that reflects the unit topic. Our group's name: _____

Consider a 55-cm-long journal bearing that is lubricated with oil whose viscosity is $0.1 \mathrm{kg} / \mathrm{m} \cdot \mathrm{s} \text { at } 20^{\circ} \mathrm{C}$ at the beginning of operation and $0.008 kg/m \cdot s$ at the anticipated steady operating temperature of $80^{\circ} \mathrm{C}.$ The diameter of the shaft is 8 cm, and the average gap between the shaft and the journal is 0.08 cm. Determine the torque needed to overcome the bearing friction initially and during steady operation when the shaft is rotated at 1500 rpm.

The polar curve $r=f(\theta),(\alpha \leq \theta \leq \beta)$, can be parametrized: $x=r \cos \theta=f(\theta) \cos \theta$, $y=r \sin \theta=f(t) \sin \theta$ Derive the formula for the arc length element for the polar curve from that for a parametric curve.

When 4.7 is multiplied by a negative power of 10, how does the new position of the decimal point relate to the negative exponent?

How are real numbers different from irrational numbers?

Graph the curve C traced out by each vector function and show its orientation. $\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}$

Explain what is wrong with the statement.

At time $t = 0$, a roast is taken out of a $40^{\circ}\mathrm{F}$ refrigerator and put in a $350^{\circ}\mathrm{F}$ oven. If $H$ represents the temperature of the roast at time $t$ minutes after it is put in the oven, we have $dH/dt = k(H − 40).$

Explain what is wrong with the statement. At time t = 0, a roast is taken out of a $40^{\circ} \mathrm{F}$ refrigerator and put in a $350^{\circ} \mathrm{F}$ oven. If H represents the temperature of the roast at time t minutes after it is put in the oven, we have dH/dt = k(H - 40).

13. Analyze Relationships The figure shown TN 10 in. . 9; : i: ' Bin. is made up of a triangle and a square. The 4 perimeter of the figure is 56 inches. What is r oe"? the area of the figure? Explain.

Find the area of the parallelogram with one vertex at P and sides PQ and PR. P=(-4,6,3); Q=(1,1,-5); R=(2,2,2)

A homeowner buys a new refrigerator with no freezer compartment and a deep freezer for the new kitchen. Which of these devices would you expect to have a lower COP? Why?

A horizontal 5-cm-diameter water jet with a velocity of 18 m/s impinges normally upon a vertical plate of mass 1000 kg. The plate rides on a nearly frictionless track and is initially stationary. When the jet strikes the plate, the plate begins to move in the direction of the jet. The water always splatters in the plane of the retreating plate. Determine (a) the acceleration of the plate when the jet first strikes it (time = 0), (b) the time it takes for the plate to reach a velocity of 9 m/s, and (c) the plate velocity 20 s after the jet first strikes the plate. For simplicity, assume the velocity of the jet is increased as the cart moves such that the impulse force exerted by the water jet on the plate remains constant.

Find the area of the parallelogram with one vertex at P and sides PQ and PR. P=(0,1,1); Q=(2,0,-4); R=(-3,-2,1)