Math 3 Quiz: Sensitivity Analysis

A bridge's load capacity $L$ (in tons) varies with temperature $T$ (°F) according to $L = 2000 - 0.5T + 0.001T^2$. During a heat wave, the temperature rises from 75°F to 105°F. If the bridge currently supports 80% of its capacity at 75°F, what safety margin remains at the higher temperature?

A manufacturing process has a yield rate $Y = 0.95 - 0.002T^{1.5}$, where $T$ is the operating temperature in degrees above room temperature. If an equipment malfunction causes the temperature to fluctuate 8°C higher than the planned 25°C above room temperature, what is the approximate change in yield rate?

A company's quarterly profit $P$ (in thousands of dollars) is modeled by $P = 50x - 2x^2 - 15$, where $x$ represents the number of products sold (in hundreds). If the company currently sells 800 products and is considering increasing sales by 10%, what is the approximate change in quarterly profit?

An environmental scientist models the pH level of a lake as $pH = 7.2 - 0.8\log(x)$ where $x$ is the concentration of pollutants in parts per million (ppm). If pollution increases from 10 ppm to 15 ppm, what is the approximate change in pH level?

A projectile's height $h$ (in feet) after $t$ seconds is given by $h(t) = -16t^2 + 64t + 80$. If the initial velocity coefficient changes from 64 to 60 ft/s while keeping other parameters constant, how does this affect the maximum height achieved?

A population growth model predicts that the number of bacteria $N$ after $t$ hours is $N(t) = 1000 \cdot 2^{0.3t}$. If the growth rate parameter changes from 0.3 to 0.35, what is the approximate percent increase in population size after 8 hours?

A logistics company models delivery time $T$ (in hours) as a function of distance $d$ (in miles) and traffic density $f$ using $T(d,f) = 0.02d + 3\sqrt{f}$. Currently, a route has $d = 150$ miles and traffic density $f = 16$. If traffic density increases to $f = 25$ while distance remains constant, how does this change affect delivery time?