The acidity of a solution is measured by its pH, with lower pH values indicating more acidity. A study of acid rain was undertaken in Colorado between 1975 and 1978, in which the acidity of rain was measured for 150 consecutive weeks. The data followed a generally linear pattern. If P is the pH and t is the number of weeks into the study, the regression line was determined to be $P=5.43-0.0053 t$. According to the line , what was the pH at the begin of the study? At the end of the study (t=150)?

Firefighting equipment requires that the exit velocity of the firehose be 30 m/s at an elevation of 45 m above the hydrant. The nozzle at the end of the hose has a contraction ratio of 4:1 (Ae/Ahose = 1/4). The head loss in the hose is 8V2/2g, where V is the velocity in the hose. What must the pressure be at the hydrant to meet this requirement? The pipe supplying the hydrant is much larger than the firehose.

Use the Poincare-Bendixson Theorem to prove the existence of a nontrivial periodic solution of each of the following differential equations. $\ddot{z}+\left(z^{2}+\dot{z}^{4}-2\right) z=0$

Find a solution u(x, t) of the following problems.

$$\frac{\partial u}{\partial t}=1.71 \frac{\partial^{2} u}{\partial x^{2}} ; \quad\left\{\begin{array}{l} {u(x, 0)=\sin \pi x / 2+3 \sin 5 \pi x / 2, \quad 0<x<2} \\ {u(0, t)=u(2, t)=0} \end{array}\right.$$

A rotameter is used to measure the flow rate of a gas with a density of 1.0 kg/m3. The scale on the rotameter indicates 5 liters/s. However, the rotameter is calibrated for a gas with a density of 1.2 kg/m3. What is the actual flow rate of the gas (in liters per second)?

(a) express $A- 1, A^2$ and all higher powers of A as a linear combination of Z and A. (The Cayley-Hamilton theorem can be of help.) (b) Calculate $e^{tA}.$

$$A=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right]$$

Consider an idealized bucket on an impulse turbine that turns the water through 180$^\circ$. Prove that the bucket speed should be one-half the incoming jet speed for a maximum power production. (Hint: Set up the momentum equation to solve for the force on the bucket in terms of Vj and Vbucket; then the power will be given by this force times Vbucket. You can use your mathematical talent to complete the problem.)

An emf of 24.0 mV is induced in a 500-turn coil when the current is changing at the rate of 10.0 A/s. What is the magnetic flux through each turn of the coil at an instant when the current is 4.00 A?

For the two-dimensional eigenvalue problem $\nabla^{2} \phi=-\lambda \phi$ with $\phi=0$ on the boundary, show that eigenfunctions corresponding to different eigenvalues are orthogonal using Green’s formula.

A sphere 8 inches in diameter falls at 4 ft /s downward axially through water in a 1-ft -diameter container. Find the upward speed of the water with respect to the container wall at the midsection of the sphere.

A consumer group is interested in estimating the proportion of packages of ground beef sold at a particular store that have an actual fat content exceeding the fat content stated on the label. How many packages of ground beef should be tested to estimate this proportion to within .05 with $95\%$ confidence?

Firehoses are fitted with special nozzles. Use the Internet or contact your local fi re department to find information on operational conditions and typical hose and nozzle sizes used.

Long-term space missions require reclamation of the oxygen in the carbon dioxide exhaled by the crew. In one method of reclamation, 1.00 mol of carbon dioxide produces 1.00 mol of oxygen and 1.00 mol of methane as a byproduct. The methane is stored in a tank under pressure and is available to control the attitude of the spacecraft by controlled venting. A single astronaut exhales 1.09 kg of carbon dioxide each day. If the methane generated in the respiration recycling of three astronauts during one week of flight is stored in an originally empty 150-L tank at –45.0 $^ { \circ } \mathrm { C }$, what is the final pressure in the tank?

A car of mass 2000 kg moving with a speed of 20.0 m/s collides and locks together with a 1500-kg car at rest at a stop sign. Show that momentum is conserved in a reference frame moving at 10.0 m/s in the direction of the moving car.