Applying Radical Functions to Real-World Situations

\begin{gathered} w_\text{mouse} = 20\text{ g} = 0.02\text{ kg} \end{gathered} Now, substitute w=0.02 into R(w) to find the metabolic rate of a mouse. Next, Kriz's metabolic rate will be calculated. To do so, substitute w=81 into R(w). The n^\text{th} Roots of n^\text{th} Powers Property will be used to evaluate the rate. The same process can be used to calculate the metabolic rate of the horse. R( 625) = 70sqrt(625^3) = 8750 Finally, the results can be summarized in a table.

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

R_1 = 64 R_2 To determine the number of times the animal is heavier, the given formula for the metabolic rate will be substituted into the equation. R_1 = 64 R_2 ⇓ 70sqrt(w_1^3) = 64( 70sqrt(w_2^3) ) Now, solve the obtained equation for w_1. This means that an animal that metabolizes 64 times more calories per day is 256 times heavier.

Both sides of the equation can now be squared to solve for h. It has been calculated that h = -16t^2 + vt + s. This means that the height of the ball is indeed modeled by a quadratic function. h = -16t^2 + vt + s ⇓ h = -16t^2 + 32t + 6 Now, substitute t=2 into the obtained quadratic equation to determine the desired height. After 2 seconds, the ball will be at a height of 6 feet. This is the height at which the ball was initially thrown. Now, the maximum height of the ball will be found. Notice that the function is already written in standard form. h = -16t^2 + 32t + 6 To calculate the maximum value of the function, start by determining the x-coordinate of the vertex of the corresponding of parabola. The x-coordinate of the vertex is 1. The maximum value h_\text{max} can be calculated by substituting t=1 into the function rule. Finally, the difference between the maximum height and the height after 2 seconds will be calculated. \begin{gathered} h_\text{max} - h = 22 - 6 = 16 \end{gathered}

t > 0 ⇒ 1/5t > 0 This means that Magdalena started the race in the first group. In Kriz's case, the distance covered is given by a square root function. Since the distance k(t) = sqrt(2t-16) cannot be an imaginary number, this function is defined for values of t such that the radicand is non-negative. 2t-16 ≥ 0 ⇔ t ≥ 8 Because the function is not defined for values of t less than 8, Kriz must not have started the race yet. Moreover, the square root of a positive number is also positive. t > 8 ⇒ sqrt(2t-16) > 0 This means that Kriz started running 8 minutes after Magdalena, at t=8. Consequently, the delay was 8 minutes.

m(t) &= 15t k(t) &= sqrt(2t-16) Any times at which Magdalena and Kriz ran side by side will be given by the values of t such that their covered distances are equal. m(t) = k(t) ⇓ 1/5t = sqrt(2t-16) The combined equation can be solved by graphing. Start with the linear function. To do so, the function rule must be written in slope-intercept form. Then, the slope m and y-intercept b can be easily identified. y = mt + b ⇓ y = 1/5t + 0 Now, the obtained equation can be graphed by plotting its y-intercept. Then, the slope will be used to determine another point that satisfies the equation, and the points will be connected with a line.

Next, the second equation y=sqrt(2t-16) will be graphed in the same coordinate plane by using a table of values. Remember to substitute t ≥ 8 so that the square root can be calculated right away.

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

All these points will be plotted and connected with a smooth curve.

According to the graph, Magdalena ran the first 2 kilometers in 10 minutes, while Kriz ran the first 2 kilometers in 2 minutes. They met 10 minutes from the start of the marathon. After this time, Kriz was in front of Magdalena for a while. t = 10 ⇒ m(t) = k(t) However, Kriz's rhythm was declining while Magdalena's remained constant. Then, 40 minutes after the start of the race, Magdalena caught up with Kriz and from then on, she took the lead until the end of the marathon. Nevertheless, notice that both ran the marathon in 50 minutes. t = 40 ⇒ m(t) = k(t) It can be concluded that the distances covered by Magdalena and Kriz were equal at t=10 and t=40. This means that they ran side by side 10 minutes and 40 minutes after the start of the marathon.

L = 3 + x To write the desired function, substitute 3+x for L into the given function for the period of a pendulum. T = 2π sqrt(L/g) ⇓ T = 2π sqrt(3+x/g) Next, substitute g=32 and π ≈ 3.14 to simplify the function rule.

3 + x ≥ 0 ⇕ x ≥ -3 Therefore, the domain of the function is all real numbers greater than or equal to -3. Next, make a table of values to graph the function. Substitute values of x so that the square root can be easily calculated.

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

Next, the points from the table will be plotted on a coordinate plane and connected with a smooth curve.

Since the square root of a greater number is also greater, the function will increase to infinity. Additionally, the square root of a non-negative number is also non-negative. Consequently, the range of the function consists of numbers greater than or equal to 0. The obtained domain and range correspond to option C.

It was determined that the range of this function consists of real numbers greater than or equal to 0. Also, because the value of the function keeps increasing without bound, the end behavior of the function is up. As x → +∞, T → +∞ Moreover, the duration of the period of the pendulum increases more and more slowly as the value of x increases. Note that this relationship goes both ways. This means that the longer the period is, the longer the pendulum that creates it needs to be.

y &= sqrt(t) [0.3em] P_1(t) &= 1/2sqrt(t - 2) + 2 First, notice that 2 is subtracted from the rule's input t. This means that the graph of the function is horizontally translated to the right by 2 units. Next, notice that the cube root in the function rule is multiplied by a positive number that is less than 1. This means that the obtained graph should now be shrunk vertically by a factor of 12. To finish the function rule of P_1, 2 is added to 12sqrt(t - 2). This implies that the corresponding graph will be vertically translated up 2 units. Finally, all of the applied transformations will be concluded.

• Horizontal translation 2 units right.
• Vertical shrink by a factor of 12.
• Vertical translation 2 units up.

Keep in mind that the order of the transformations may differ. For example, the horizontal translation could be the last transformation.

Average Rate of Change = f(x_2)-f(x_1)/x_2-x_1 To find the average rate of change of P_1 over the intervals [2, 10] and [10, 18], first, the values P_1(2), P_1(10), and P_1(18) will be calculated in a table of values. Use a calculator if necessary.

t	1/2sqrt(t-2) + 2	P_1(t)
2	1/2sqrt(2-2) + 2	2
10	1/2sqrt(10-2) + 2	3
18	1/2sqrt(18-2) + 2	≈ 3.26

Now, substitute values of t and P_1(t) into the formula for the average rate of change over both intervals.

Interval [t_1, t_2]	P_1(t_2)-P_1(t_1)/t_2-t_1	Substitute and Evaluate
[ 2, 10]	P_1( 10)-P_1( 2)/10- 2	3-2/10-2 = 0.125
[ 10, 18]	P_1( 18)-P_1( 10)/18- 10	3.26-3/18-10 ≈ 0.033

The average rate of change R_1 is greater than R_2. Also, both values are positive. This means that after 10 years, the profit of the first company will increase but more slowly.

t	1/2sqrt(t-2) + 2	P_1(t)
0	1/2sqrt(0-2) + 2	≈ 1.37
10	1/2sqrt(10-2) + 2	3

Next, approximate the values of P_2(0) and P_2(10) from the graph of the profit P_2.

It was found that P_2(0) is about 1.25 and P_2(10) is equal to 3. Now, both average rates of change over the interval [ 0, 10] can be determined. Use a calculator if needed.

Profit Function	f(x_2)-f(x_1)/x_2-x_1	Substitute and Evaluate
P_1	P_1( 10)-P_1( 0)/10- 0	3-1.37/10-0 = 0.163
P_2	P_2( 10)-P_2( 0)/10- 0	3-1.25/10-0 = 0.175

The average rate of change of the annual profit of the second company is greater over the next 10 years. This means that the average increase in profit of the second company is greater. Keep in mind that the average rate of change measures the average change in profit, not the actual profit.