Applying Radical Functions to Real-World Situations

For every second, let's visualize the circular waves generated by the thrown stone.

As time t increases, the radius of a circle increases at a rate of 25 centimeters per second. To find the radius r at any second, multiply the rate by that time. r=25t We can also think of this as calculating the distance in every direction given the speed of the wave. The area A of any circle is given by π times the square of its radius. A=π r^2 This equation represents how the area of the circle depends on the radius. To represent the relationship between area and time, we will bring the obtained equations together to perform a composition of functions. Both equations have the radius as a variable. We can substitute the first equation into the second one.

Start by noticing that the area is given in square feet while the formula obtained in Part A was calculated for centimeters. We can use a conversion factor to rewrite the given area in square centimeters. Recall that 1 foot is equal to 30.48 centimeters. 200 ft^2 = 200( 1ft*30.48cm/1ft )^2 ⇕ 200 ft^2 = 185 806.08 cm^2 Finally, let's substitute A=185 806.08 into the equation obtained in Part A and solve for t.

This means that it will take the wave about 9.73 seconds to hit the bounding stones of the pond.

Mammal	Mass
Mouse	$20$ g
Kriz	$81$ kg
Horse	$625$ kg

Mammal	Mass $(kg)$	Metabolic Rate $(kcal / day)$
Mouse	$0.02$	$\approx 3.72$
Kriz	$81$	$1890$
Horse	$625$	$8750$

$t$	$2 t - 16$	$y = 2 t - 16$
$8$	$2 (8) - 16$	$0 = 0$
$10$	$2 (10) - 16$	$4 = 2$
$16$	$2 (16) - 16$	$16 = 4$
$26$	$2 (26) - 16$	$36 = 6$
$40$	$2 (40) - 16$	$64 = 8$
$58$	$2 (58) - 16$	$100 = 10$

$x$	$1.11 3 + x$	$T = 1.11 3 + x$
$- 3$	$1.11 3 + (- 3)$	$0$
$- 2$	$1.11 3 + (- 2)$	$1.11$
$1$	$1.11 3 + 1$	$2.22$
$6$	$1.11 3 + 6$	$3.33$
$13$	$1.11 3 + 13$	$4.44$

Option	Statement
A	The average increase in profit of the first company will be greater over the next $10$ years.
B	The average increase in profit of the second company will be greater over the next $10$ years.
C	On average, the profit of the companies increases at the same rate.
D	On average, the annual profit of the second company is about $0.175$ millions of dollars.

Applying Radical Functions to Real-World Situations

Catch-Up and Review

How to Calculate the Speed of a Car?

Kleiber's Law

Hint

Solution

A Ball Thrown Vertically

Hint

Solution

A Quarter Marathon

Hint

Solution

Period of a Pendulum

Hint

Solution

Profit of Software Start-ups

Hint

Solution

Comparing Different Ways of Calculating the Speed of a Car

Hint

Solution

Applying Radical Functions to Real-World Situations

Recommended exercises

	8 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

$t$	$\frac{1}{2} 3 t - 2 + 2$	$P_{1} (t)$
$2$	$\frac{1}{2} 3 2 - 2 + 2$	$2$
$10$	$\frac{1}{2} 3 10 - 2 + 2$	$3$
$18$	$\frac{1}{2} 3 18 - 2 + 2$	$\approx 3.26$

Interval $[t_{1}, t_{2}]$	$\frac{P _{1} ( t _{2} ) - P _{1} ( t _{1} )}{t _{2} - t _{1}}$	Substitute and Evaluate
$[2, 10]$	$\frac{P _{1} ( 1 0 ) - P _{1} ( 2 )}{1 0 - 2}$	$\frac{3 - 2}{1 0 - 2} = 0.125$
$[10, 18]$	$\frac{P _{1} ( 1 8 ) - P _{1} ( 1 0 )}{1 8 - 1 0}$	$\frac{3 . 2 6 - 3}{1 8 - 1 0} \approx 0.033$

$t$	$\frac{1}{2} 3 t - 2 + 2$	$P_{1} (t)$
$0$	$\frac{1}{2} 3 0 - 2 + 2$	$\approx 1.37$
$10$	$\frac{1}{2} 3 10 - 2 + 2$	$3$

Profit Function	$\frac{f ( x _{2} ) - f ( x _{1} )}{x _{2} - x _{1}}$	Substitute and Evaluate
$P_{1}$	$\frac{P _{1} ( 1 0 ) - P _{1} ( 0 )}{1 0 - 0}$	$\frac{3 - 1 . 3 7}{1 0 - 0} = 0.163$
$P_{2}$	$\frac{P _{2} ( 1 0 ) - P _{2} ( 0 )}{1 0 - 0}$	$\frac{3 - 1 . 2 5}{1 0 - 0} = 0.175$

Input	$18 d$	$v_{1} (d)$	$16 3 p$	$v_{2} (p)$
$0$	$18 (0)$	$0$	$1630$	$0$
$216$	$18 (216)$	$\approx 62.35$	$163216$	$96$

Function	$\frac{f ( x _{2} ) - f ( x _{1} )}{x _{2} - x _{1}}$	Substitute and Evaluate
$v_{1}$	$\frac{v _{1} ( 2 1 6 ) - v _{1} ( 0 )}{2 1 6 - 0}$	$\frac{6 2 . 3 5 - 0}{2 1 6 - 0} \approx 0.29$
$v_{2}$	$\frac{v _{2} ( 2 1 6 ) - v _{2} ( 0 )}{2 1 6 - 0}$	$\frac{9 6 - 0}{2 1 6 - 0} \approx 0.44$

$d$	$18 d$	$v_{1} (d)$
$0$	$18 (0)$	$0$
$2$	$18 (2)$	$6$
$4$	$18 (4)$	$\approx 8.5$
$6$	$18 (6)$	$\approx 10.4$
$8$	$18 (8)$	$12$
$10$	$18 (10)$	$\approx 13.4$

$p$	$16 3 p$	$v_{2} (p)$
$- 8$	$16 3 - 8$	$- 32$
$- 6$	$16 3 - 6$	$\approx - 29.1$
$- 4$	$16 3 - 4$	$\approx - 25.4$
$- 2$	$16 3 - 2$	$\approx - 20.2$
$0$	$1630$	$0$
$2$	$1632$	$\approx 20.2$
$4$	$1634$	$\approx 25.4$
$6$	$1636$	$\approx 29.1$
$8$	$1638$	$32$

Metabolic Rate
The rate at which a person or an animal burns calories to maintain its body weight.