4.7 Applied optimization problems  (Page 7/8)

A patient’s pulse measures $\text{70 bpm, 80 bpm, then 120 bpm}.$ To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression ${\left(x-70\right)}^2+{\left(x-80\right)}^2+{\left(x-120\right)}^2?$ What value minimizes it?

In the previous problem, assume the patient was nervous during the third measurement, so we only weight that value half as much as the others. What is the value that minimizes ${\left(x-70\right)}^2+{\left(x-80\right)}^2+\frac{1}{2}{\left(x-120\right)}^2?$

You can run at a speed of $6$ mph and swim at a speed of $3$ mph and are located on the shore, $4$ miles east of an island that is $1$ mile north of the shoreline. How far should you run west to minimize the time needed to reach the island?

Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, $\theta .$

Find at what angle $\theta$ the lifeguard should swim to reach the drowning person in the least amount of time.

A truck uses gas as $g\left(v\right)=av+\frac{b}{v},$ where $v$ represents the speed of the truck and $g$ represents the gallons of fuel per mile. At what speed is fuel consumption minimized?

Find the cost per mile at speed $v.$

Find the cheapest driving speed.

Find the profit function for the number of pizzas. How many pizzas gives the largest profit per pizza?

Assume that $R\left(x\right)=10x$ and $C\left(x\right)=2x+x^2.$ How many pizzas sold maximizes the profit?

Assume that $R\left(x\right)=15x,$ and $C\left(x\right)=60+3x+\frac{1}{2}{x}^2.$ How many pizzas sold maximizes the profit?

Choose $x$ to maximize the sum of their areas.

Choose $x$ to minimize the sum of their areas.

$xy$

$x^2{y}^2$

$y-\frac{1}{x}$

$x^2-y$

Find the volume of the largest right circular cylinder that fits in a sphere of radius $1.$

Find the volume of the largest right cone that fits in a sphere of radius $1.$

Find the area of the largest rectangle that fits into the triangle with sides $x=0,y=0$ and $\frac{x}{4}+\frac{y}{6}=1.$

Find the largest volume of a cylinder that fits into a cone that has base radius $R$ and height $h.$

Find the dimensions of the closed cylinder volume $V=16\pi$ that has the least amount of surface area.

Find the dimensions of a right cone with surface area $S=4\pi$ that has the largest volume.

[T] Where is the line $y=5-2x$ closest to the origin?

[T] Where is the line $y=5-2x$ closest to point $\left(1,1\right)?$

[T] Where is the parabola $y=x^2$ closest to point $\left(2,0\right)?$

[T] Where is the parabola $y=x^2$ closest to point $\left(0,3\right)?$

A window is composed of a semicircle placed on top of a rectangle. If you have $20\text{ft}$ of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use $r$ to represent the radius of the semicircle.

You have a garden row of $20$ watermelon plants that produce an average of $30$ watermelons apiece. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. How many extra watermelon plants should you plant?

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $\text{\$}5\text{/}{\text{ft}}^2$ and the material for the sides costs $\text{\$}2\text{/}{\text{ft}}^2.$ You need a box with volume $4{\text{ft}}^2.$ Find the dimensions of the box that minimize cost. Use $x$ to represent the length of the side of the box.

You are building five identical pens adjacent to each other with a total area of $1000{\text{m}}^2,$ as shown in the following figure. What dimensions should you use to minimize the amount of fencing?

You are the manager of an apartment complex with $50$ units. When you set rent at $\text{\$}800\text{/}\text{month,}$ all apartments are rented. As you increase rent by $\text{\$}25\text{/}\text{month,}$ one fewer apartment is rented. Maintenance costs run $\text{\$}50\text{/}\text{month}$ for each occupied unit. What is the rent that maximizes the total amount of profit?