use Archimedes’ Principle, which states that when a sphere of radius r with density

$$d_s$$

is placed in a liquid of density

$$d_L=62.5lb/ft^3$$

,it will sink to a depth h where .

$$\dfrac{π}{3}(3rh^2-h^3)d_L=\dfrac{4}{3}πr^3d_s$$

Find an approximate value for h if:

$$r=5 ft$$

and

$$d_s=20lb/ft^3$$

The perimeter of a soccer field is 300 yards. If the length is 50 yards longer than the width, what are the field's dimensions?

Hydrogen peroxide and the iodide ion react in acidic solution as follows: $\mathrm { H } _ { 2 } \mathrm { O } _ { 2 } ( a q ) + 3 \mathrm { I } ^ { - } ( a q ) + 2 \mathrm { H } ^ { + } ( a q ) \longrightarrow \mathrm { I } _ { 3 } ^ { - } ( a q ) + 2 \mathrm { H } _ { 2 } \mathrm { O } ( l )$ The kinetics of this reaction were studied by following the decay of the concentration of $\mathrm { H } _ { 2 } \mathrm { O } _ { 2 }$ and constructing plots of $\ln \left[ \mathrm { H } _ { 2 } \mathrm { O } _ { 2 } \right]$ versus time. All the plots were linear and all solutions had $\left[ \mathrm { H } _ { 2 } \mathrm { O } _ { 2 } \right] _ { 0 } = 8.0 \times 10 ^ { - 4 } \mathrm { mol } / \mathrm { L } .$ The slopes of these straight lines depended on the initial concentrations of $\mathrm { I } ^ { - }$ and $\mathrm { H } ^ { + }.$ The results follow:

$$\begin{matrix} \text{$\left[ \mathrm { I } ^ { - } \right] _ { 0 } ( \mathrm { mol } / \mathrm { L } )$} & \text{$\left[ \mathrm { H } ^ { + } \right] _ { 0 } ( \mathrm { mol } / \mathrm { L } )$} & \text{Slope $\left( \mathrm {min} ^ { - 1 } \right)$}\\ \hline \text{0.1000} & \text{0.0400} & \text{$- 0.120$}\\ \text{0.3000} & \text{0.0400} & \text{$- 0.360$}\\ \text{0.4000} & \text{0.0400} & \text{$- 0.480$}\\ \text{0.0750} & \text{0.0200} & \text{$- 0.0760$}\\ \text{0.0750} & \text{0.0800} & \text{$- 0.118$}\\ \text{0.0750} & \text{0.1600} & \text{$- 0.174$}\\ \end{matrix}$$

The rate law for this reaction has the form $\mathrm {Rate} = \frac { - d \left[ \mathrm { H } _ { 2 } \mathrm { O } _ { 2 } \right] } { d t } = \left( k _ { 1 } + k _ { 2 } \left[ \mathrm { H } ^ { + } \right] \right) \left[ \mathrm { I } ^ { - } \right] ^ { m } \left[ \mathrm { H } _ { 2 } \mathrm { O } _ { 2 } \right] ^ { n }$ Calculate the values of the rate constants $k _ { 1 }$ and $k _ { 2 }.$

Find the midpoint of the line segment with endpoints at the given coordinates. Then find the distance between the points.

$$( - 7.54,3.89 ) , ( 4.04 , - 0.38 )$$

Solve the literal equation for y.

$$8 x - 3 = 5 + 4 y$$

The graph of quadratic functions may have one maximum or one minimum point. The maximum point of a graph is the point with the greatest y-value coordinate. The minimum point is the point with the least y-value coordinate. Graph the equation. Find the coordinates of the point. the maximum point of the graph of

$$y = - x ^ { 2 } + 7$$

h is related to one of the parent functions described in this chapter. Identify the parent function f.

$$h(x) = -1/3 x^3$$

Find the coordinates of the midpoint of the line segment between the given points. $(0,5,-8),(4,1,-6)$

The percent price increase p on a variety of retail food prices over a 1-year period varied from 5% to 10% in all cases. Because of the distribution of p values, the assumed probability distribution for the next year is

$$f ( X ) = 2 X \quad 0 \leq X \leq 1$$

where

$$X = \left\{ \begin{array} { l l } { 0 } & { \text { when } p = 5 \% } \\ { 1 } & { \text { when } p = 10 \% } \end{array} \right.$$

For a continuous variable the cumulative distribution F(X) is the integral of f(X) over the same range of the variable. In this case

$$F ( X ) = X ^ { 2 } \quad 0 \leq X \leq 1$$

(a) Graphically assign RNs to the cumulative distribution, and take a sample of size 30 for the variable. Transform the X values into interest rates. (b) Calculate the average p value for the sample.

The occurrence of random events (such as phone calls or e-mail messages) is often idealized using an exponential distribution. If $\lambda$ is the average rate of occurrence of such an event, assumed to be constant over time, then the average time between occurrences is $\lambda ^ { - 1 }$ (for example, if phone calls arrive at a rate of $\lambda = 2 / \mathrm { min }$, then the mean time between phone calls is $\lambda ^ { - 1 } = \frac { 1 } { 2 }$ min). The exponential distribution is given by $f(t)=\lambda e ^ { - \lambda t }$, for $0 \leq t \lt \infty$.

Suppose you work at a customer service desk and phone calls arrive at an average rate of $\lambda _ { 1 } = 0.8 / \min$ (meaning the average time between phone calls is 1/0.8=1.25 min). The probability that a phone call arrives during the interval [0, T] is $p(t)=\int _ { 0 } ^ { T } \lambda _ { 1 } e ^ { - \lambda _ { 1 } t } d t$. Find the probability that a phone call arrives during the first 45 s (0.75 min) that you work at the desk.

Compare the hybridization of Si atoms in Si(s) with that of C atoms in graphite. If silicon were to adopt the graphite structure, would its electrical conductivity be high or low?

Determine whether the set

$$\mathcal { B }$$

is a basis for the vector space V.

$$V = M _ { 22 } , \mathcal { B } = \left\{ \left[ \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right] , \left[ \begin{array} { l l } { 1 } & { 1 } \\ { 1 } & { 0 } \end{array} \right] , \left[ \begin{array} { r r } { 1 } & { - 1 } \\ { - 1 } & { 1 } \end{array} \right] \right\}$$

$\mathbf{v}_{1}=\langle 4,6\rangle, \mathbf{v}_{2}=\langle- 3,-6\rangle, \mathbf{v}_{3}=\langle- 8,4\rangle,$ and $\mathbf{v}_{4}=\langle 10,15\rangle$ Which two vectors are parallel?

Find the missing terms in the arithmetic sequence.

$$9 , \square , \square ,\square, 37$$