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A triangular tank measuring 12m tall, 15m long, and 4m across contains water 6m deep. Which of the following integrals represents the work required to pump all the water out of the top of the tank? (Assume the acceleration due to gravity is 10 m/s\(^2\) and the density of water is 100 kg/m\(^3\).)  

The table below shows values of a function \(f(x)\).  Use Trapezoid Rule with four equal subintervals (\(n=4\)) to estimate the value of the integral \[\displaystyle \int_1^5 f(x) \; dx.\] 

Uranium 238 is a radioactive material with applications in nuclear technology. It decays exponentially with a half-life of about 4.5 billion years. If given an initial mass of 5 grams, find the mass that remains after 3 billion years. Note: \(\exp(x)\) means \(e^x\) so for example, \(\exp(5) = e^5.\)  

Suppose the cost of utilities for a house is given by \[C(t) = 50 \cos\left(\frac{\pi}{6}t\right) + 100,\] where \(t\) is in months. Find the average cost of utilities from December (\(t=0\)) to June (\(t=6\)).

Evaluate the integral, if it exists.\[\int_1^\infty \frac{4}{x^3} \;dx\]

Find the equation of the tangent line to the curve \[y = 3 \arctan(2x)\] at the point \(\displaystyle x = \frac{1}{2}\).

Find the total mass of the \(1\) m rod whose linear density function is given by \[\rho(x) = 4x(1+x^2)^3 \;\;\text{    kg/m}\] for \(0 \leq x \leq 1\).

Order: Omeprazole 11mg, gastrostomy tube, every 12 hours, for reflux  Child’s weight: 22 kg Safe dose range: 1-1.5mg/kg/day in two divided doses   a) What is the safe low dose (mg/dose) for this child? b) What is the safe high dose (mg/dose) for this child?  c) Is the ordered dose safe for this child?

Which of the following is the partial fractions decomposition of the rational function \[f(x)=\frac{5x^2+1}{(x^2-1)^2(x^2+9)}\;\;?\]

Order: Vancomycin 600mg, intravenously, every 6 hours, for moderate infection  Child’s weight: 50 kg  Safe dose range: 10-15mg/kg/dose (max dose 750mg) every 6 hours    a) What is the safe low dose (mg/dose) for this child?  b) What is the safe high dose (mg/dose) for this child? c) Is the ordered dose safe for this child?