8.7 Applications  (Page 4/4)

An inlet pipe can fill a tank in $a$ units of time. An outlet pipe can empty the tank in $b$ units of time. If both pipes are open, how many units of time are required to fill the tank? Are there any restrictions on $a$ and $b$ ?

A delivery boy, working alone, can deliver all his goods in 6 hours. Another delivery boy, working alone, can deliver the same goods in 5 hours. How long will it take the boys to deliver all the goods working together?

A Space Shuttle astronaut can perform a certain experiment in 2 hours. Another Space Shuttle astronaut who is not as familiar with the experiment can perform it in $2\frac{1}{2}$ hours. Working together, how long will it take both astronauts to perform the experiment?

One person can complete a task 8 hours sooner than another person. Working together, both people can perform the task in 3 hours. How many hours does it take each person to complete the task working alone?

Find two consecutive integers such that two thirds of the smaller number added to the other yields 11.

Find two consecutive integers such that three fourths of the smaller number added to the other yields 29.

The width of a rectangle is $\frac{2}{5}$ its length. Find the dimensions if the perimeter is 42 meters.

The width of a rectangle is $\frac{3}{7}$ the length. Find the dimensions if the perimeter is 60 feet.

Two sides of a triangle have the same length. The third side is twice as long as either of the other two sides. The perimeter of the triangle is 56 inches. What is the length of each side?

In a triangle, the second side is 3 inches longer than first side. The third side is $\frac{3}{4}$ the length of the second side. If the perimeter is 30 inches, how long is each side?

The pressure due to surface tension in a spherical drop of liquid is given by $P=\frac{2T}{r},$ where $T$ is the surface tension of the liquid and $r$ is the radius of the drop. If the liquid is a bubble, it has two surfaces and the surface tension is given by

$$P=\frac{2T}{r}+\frac{2T}{r}=\frac{4T}{r}$$
(a) Determine the pressure due to surface tension within a soap bubble of radius 2 inches and surface tension 28.
(b) Determine the radius of a bubble if the pressure due to surface tension is 52 and the surface tension is 39.

The equation $\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$ relates the distance $p$ of an object from a lens and the image distance $q$ from the lens to the focal length $f$ of the lens.

(a) Determine the focal length of a lens in which an object 10 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 6 inches and the image distance is 10 inches.
(c) Determine how far an image will be from a lens that has a focal length of $4\frac{4}{5}$ cm and the object is 12 cm away from the lens.

Person A can complete a task in 4 hours, person B can complete the task in 6 hours, and person C can complete the task in 3 hours. If all three people are working together, how long will it take to complete the task?

Three inlet pipes can fill a storage tank in 4, 6, and 8 hours, respectively. How long will it take all three pipes to fill the tank?

An inlet pipe can fill a tank in 10 hours. The tank has two drain pipes, each of which can empty the tank in 30 hours. If all three pipes are open, can the tank be filled? If so, how long will it take?

An inlet pipe can fill a tank in 4 hours. The tank has three drain pipes. Two of the drain pipes can empty the tank in 12 hours, and the third can empty the tank in 20 hours. If all four pipes are open, can the tank be filled? If so, how long will it take?

( [link] ) Factor $12a^2+13a-4.$

( [link] ) Find the slope of the line passing through the points $\left(4,-3\right)$ and $\left(1,-6\right).$

( [link] ) Find the quotient: $\frac{2x^2-11x-6}{x^2-2x-24}÷\frac{2x^2-3x-2}{x^2+2x-8}.$

( [link] ) Find the difference: $\frac{x+2}{x^2+5x+6}-\frac{x+1}{x^2+4x+3}.$

( [link] ) Solve the equation $\frac{9}{2m-5}=-2.$