Chapter 1.10, Problem 75E

(a)

To determine

The plot for the given data.

Answer to Problem 75E

The sketch of a scatter plot is shown in figure.

Explanation of Solution

Given information:

The data collected of reading of the water temperatures is $\left(1000,4.2\right),\left(4000,1.2\right),\left(2000,1.9\right),$

  $\left(5000,0.9\right),\left(3000,1.4\right)$ .

Calculation:

The scatter plot of data is shown in below figure.

  

Therefore, the sketch of a scatter plot is shown in figure.

(b)

To determine

The value of $k$ for each pair of coordinates.

Answer to Problem 75E

The value of $k$ for pair $\left(1000,4\cdot 2\right)$ is $4200$ , the value of $k$ for pair $\left(2000,1\cdot 9\right)$ is , $3800$ , the value of $k$ for pair $\left(4000,1\cdot 2\right)$ is $4800$ ,the value of $k$ for pair $\left(3000,1\cdot 4\right)$ is $4200$ , the value of $k$ for pair $\left(5000,0\cdot 9\right)$ is $4500$ .

Explanation of Solution

Given information:

The data collected of reading of the water temperatures is $\left(1000,4\cdot 2\right),\left(4000,1\cdot 2\right),\left(2000,1\cdot 9\right),$

  $\left(5000,0\cdot 9\right),\left(3000,1\cdot 4\right)$ .

Calculation:

In case when the value of $d$ increasing, the value of $C$ starts decreasing or we can say that $d$ is inversely proportion to $C$ . Therefore, the model equation is $C=\frac{k}{d}$ .

Calculate the value of $k$ for pair $\left(1000,4\cdot 2\right)$ .

  $\begin{array}{c}4\cdot 2=\frac{k}{1000}\\ k=1000\left(4\cdot 2\right)\\ k=4200\end{array}$

Calculate the value of $k$ for pair $\left(2000,1\cdot 9\right)$ .

  $\begin{array}{c}1\cdot 9=\frac{k}{2000}\\ k=2000\left(1\cdot 9\right)\\ k=3800\end{array}$

Calculate the value of $k$ for pair $\left(4000,1\cdot 2\right)$

  $\begin{array}{c}1\cdot 2=\frac{k}{4000}\\ k=4000\left(1\cdot 2\right)\\ k=4800\end{array}$

Calculate the value of $k$ for pair $\left(3000,1\cdot 4\right)$ .

  $\begin{array}{c}1\cdot 4=\frac{k}{3000}\\ k=3000\left(1\cdot 4\right)\\ k=4200\end{array}$

Calculate the value of $k$ for pair $\left(5000,0\cdot 9\right)$ .

  $\begin{array}{c}4\cdot 9=\frac{k}{5000}\\ k=5000\left(4\cdot 2\right)\\ k=4500\end{array}$

Therefore, the value of $k$ for pair $\left(1000,4\cdot 2\right)$ is $4200$ , the value of $k$ for pair $\left(2000,1\cdot 9\right)$ is , $3800$ , the value of $k$ for pair $\left(4000,1\cdot 2\right)$ is $4800$ ,the value of $k$ for pair $\left(3000,1\cdot 4\right)$ is $4200$ , the value of $k$ for pair $\left(5000,0\cdot 9\right)$ is $4500$ .

(c)

To determine

The mean value of $k$ and the inverse variation model $C=\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ .

Answer to Problem 75E

The mean value of $k$ is $4300$ and the inverse variation model $C=\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ is $C=\frac{4300}{d}$ .

Explanation of Solution

Given information:

The data collected of reading of the water temperatures is $\left(1000,4\cdot 2\right),\left(4000,1\cdot 2\right),\left(2000,1\cdot 9\right),$

  $\left(5000,0\cdot 9\right),\left(3000,1\cdot 4\right)$ .

Calculation:

Calculate the mean value of $k$ .

  $\begin{array}{c}\text{mean}\text{value}=\frac{4200+3800+4200+4800+4500}{5}\\ =21500\\ =4300\end{array}$

Using the model equation is $C=\frac{k}{d}$ .

  $\begin{array}{l}C=\frac{k}{d}\\ C=\frac{4300}{d}\end{array}$

Therefore, the mean value of $k$ is $4300$ and the inverse variation model $C=\raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ is $C=\frac{4300}{d}$ .

(d)

To determine

The plot for the data points and inverse model from part (c).

Answer to Problem 75E

The plot for the data points and inverse model is shown in figure.

Explanation of Solution

Given information:

The data collected of reading of the water temperatures is $\left(1000,4\cdot 2\right),\left(4000,1\cdot 2\right),\left(2000,1\cdot 9\right),$

  $\left(5000,0\cdot 9\right),\left(3000,1\cdot 4\right)$ .

Calculation:

Using the graphing utility the plot of the data and inverse model from part (c).

  

Therefore, the plot for the data points and inverse model is shown in figure.

(e)

To determine

The depth at which the water temperature is $3°C$ .

Answer to Problem 75E

The depth at which the water temperature $3°C$ is $1433.33\text{m}$ .

Explanation of Solution

Given information:

The data collected of reading of the water temperatures is $\left(1000,4\cdot 2\right),\left(4000,1\cdot 2\right),\left(2000,1\cdot 9\right),$

  $\left(5000,0\cdot 9\right),\left(3000,1\cdot 4\right)$ .

Calculation:

Substitute the $3$ for $C$ in the model equation $C=\frac{4300}{d}$ .

  $\begin{array}{c}3=\frac{4300}{d}\\ d=\frac{4300}{3}\\ d\approx 1433.33\end{array}$

Therefore, the depth at which the water temperature $3°C$ is $1433.33\text{m}$ .