6A Formula Applications

Look at the following formula $V=\pi r^{2}h$ which gives the volume of a circular cylinder water tank. If we use 3.14 as an approximation of $\pi$ and we know that radius needs to be 5 feet, then how tall should the water tank be if we need it to hold 1570 cubic feet of water? The rule of thumb is to plug in what you know and then use equation solving to figure out what you don’t know.

Look at the formula and plug in $\pi$, r and V. Now simplify and solve for h.

$$\begin{array}{l}V=\pi r^{2}h\\ 1570=3.14{(5)}^{2}h\\ 1570=3.14(25)h\\ 1570=78.5h\end{array}$$

Now we solve for h by dividing both sides by 78.5 and we find that h is 20 feet. So the tank needs to be 20 feet tall!

$$\begin{array}{l}1570=78.5h\\ \frac{1570}{78.5}=\frac{{}^{1}78.5}{78.5}h\\ \frac{1570}{78.5}=1h\\ 20=h\end{array}$$

We used regression theory in statistics to find the profit formula $P=91.3X-643.5$ for a chain of restaurants. X represents the number of hours that their employees work and P gives the restaurants weekly profit. If the company wants to make $3000 per week in profit, how many hours a week should their employees work? Round your answer to the tenths place.

Since we know the company wants the profits P = 3000, we will plug in 3000 for P.

$$\begin{array}{l}P=91.3X-643.5\\ 3000=91.3X-643.5\end{array}$$

Now all we have to do is solve the equation for X. We can eliminate the decimals by multiplying both sides by 10.

$$\begin{array}{l}10(3000)=10(91.3X-643.5)\\ 30000=913X-6435\end{array}$$

Since the variable is already on the right side, let’s bring the constants to the opposite side by adding 6435 to both sides.

$$\begin{array}{rcl}30000& =& 913X-6435\\ +6435& & +6435\\ 36435& =& 913X\end{array}$$

Now we can divide both sides by 913 to get X by itself. We can round our answer to the tenths place so we do not need to keep dividing past the hundredths place.

$$\begin{array}{l}36435=913X\\ \frac{36435}{913}=\frac{{}^{1}913X}{913}\\ 39.90\approx X\end{array}$$

So we have found that the company should have its employees work about 39.9 hours per week. It sounds like a 40 hour work week would give them the $3000 in profit.

Try the following example with your instructor:

Practice Problems Section 6A

1. The perimeter of a rectangular vegetable garden is P = 2L + 2W. Mike wants to fence off a garden in his backyard. He needs the garden to be 47 feet long and he has 150 feet of fencing. How wide should his garden be?

   Check answer

   28 ft

2. In a previous example, a statistician used regression theory to find the profit formula $P=91.3X-643.5$ for a chain of restaurants. X represents the number of hours that their employees work each week and P gives the restaurants weekly profit. Suppose that the company needs to cut the number of hours their employees work, but they still need to make $2500 per week in profit. How many hours should they have their employees work? Round to the tenths place.

3. Look at the formula $V=\pi r^{2}h$ which gives the volume of a circular cylinder water bottle. If we use 3.14 as an approximation of $\pi$ and we know the radius needs to be 4 cm, how tall should the water bottle be if it needs to hold 502.4 cubic centimeters of water?

   Check answer

   10 cm

4. We used regression theory in statistics to find a formula that will predict the weight of a black bear based on its age. The formula was $W=65.2+2.7A$ where A is the age of the bear in months and W is the weight. Forest rangers caught a black bear in the wild and found the weight to be 194.8 pounds. How old do we predict the bear to be?

5. The volume of a box is given by the formula $V=L\times W\times H$ where V is the volume, L is the length, W is the width, and H is the height. A metal container to be used on ships needs to have a length of 28 feet and a width of 14 feet. How tall should the container be if it needs to hold 3136 cubic feet of cargo?

   Check answer

   8 ft

6. We used Regression theory in statistics to find the following formula F = 0.028C-1.98 where C is the number of calories a breakfast cereal has and F is the grams of fat the breakfast cereal has. Suppose a cereal has 1.38 grams of fat. How many calories do we expect the cereal to have?

7. The certificate of deposit formula that banks use is $A=P{(1+\frac{r}{n})}^{nT}$ where P is the principal (amount initially invested), A is the future amount in the account, T is the number of years the money is in the account, n is how many times it compounds per year (how many times they put the interest back in the account) and r is the interest rate. Lucy wants to buy a house in two years and will need a down payment of $11664. How much money (P) should she invest in her certificate of deposit if her interest rate is 8% (r = 0.08), compounded annually (n=1)?

   Check answer

   $10,000

8. We used regression theory in statistics to find the following formula F = 0.028C-1.98 where C is the number of calories a breakfast cereal has and F is the grams of fat the breakfast cereal has. Suppose a cereal has 0.82 grams of fat. How many calories do we expect the cereal to have?

9. A common formula in physics is that force = mass x acceleration ($F=m\times a$). Suppose we have a force of 13.5 newtons acting on an object which results in an acceleration of 2.5 $m/s^2$. Find the mass of the object in kilograms?

   Check answer

   5.4 kilograms

10. The surface area of a cylinder is given by the formula $S=2\pi r(h+r)$. Suppose a cylinder has a surface area of 2640 square inches and a radius of 14 inches. Find the height of the cylinder. Use $\pi \approx \frac{22}{7}$.

11. The area of a triangular region is given by the formula $A=\frac{b\times h}{2}$ where b is the base of the triangle and h is the height of the triangle. Find the height of a triangular shaped boat sail whose base is 3.5 feet and whose area is 15.75 square feet.

    Check answer

    $9$ ft

12. A common formula in physics is that force = mass $\times$ acceleration ($F=m\times a$). Suppose we have a force of 138 newtons acting on an object that has a mass of 18.4 kg. What would be the acceleration in $m/s^2$?

13. We used regression theory in statistics to find a formula that will predict the weight of a black bear based on its age. The formula was $W=65.2+2.7A$ where A is the age of the bear in months and W is the weight. How many months old do we predict a black bear to be if its weight is 119.2 pounds?

    Check answer

    20 months

14. The certificate of deposit formula that banks use is $A=P{(1+\frac{r}{n})}^{nT}$ where P is the principal (amount initially invested), A is the future amount in the account, T is the number of years the money is in the account, n is how many times it compounds per year (how many times they put the interest back in the account) and r is the interest rate. Jerry wants to save $2080.80 in 1 year. How much money should he invest (P) in his savings account if his interest rate is 4% (r = 0.04), compounded semiannually (n = 2)?

15. We used regression theory in statistics to find the following formula F = 0.028C - 1.98 where C is the number of calories a breakfast cereal has and F is the grams of fat the breakfast cereal has. Suppose a cereal has 1.94 grams of fat. How many calories do we expect the cereal to have?

    Check answer

    140 calories

16. The volume of a cone is given by the formula $V=\frac{1}{3}\pi r^{2}h$ where r is the radius of the cone and h is the height of the cone. Use the approximation that $\pi \approx \frac{22}{7}$. What is the height of a cone that has a volume of 462 cubic centimeters and a radius of 7 centimeters?

17. A statistician used regression theory to find the profit formula $P=91.3X-643.5$ for a chain of restaurants. X represents the number of hours that their employees work each week and P gives the restaurants weekly profit. Suppose that the company needs to cut the number of hours their employees work, but they still need to make $4000 per week in profit. How many hours should they have their employees work? Round to the tenths place.

    Check answer

    50.9 hours

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.