Learn Decimals with Plainmath's In-Depth Explanations and Real-World Examples

Determine ${\displaystyle z\alpha }$ for the following. (Round your answer to two decimal place)
a) ${\displaystyle \alpha =0.0089}$
b) ${\displaystyle \alpha =0.09}$
c) ${\displaystyle \alpha =0.707}$

Even though housing prices have been declining in many parts of the country, in Lara's neighborhood, they have been increasing steadily since Lara bought her home in 2006. Lara purchased her home in 2006 for $120,000, and it steadily increases its value by $4100 a year. Let V represent the market value (in dollars) of her home; x, the length of time she has lived there (in years). Complete parts a through d below.
a. Complete the following table.
$\begin{array}{|ccc|}\hline Year& x& \text{V, Market value}\\ 2006& & \$\\ 2011& & \$\\ 2014& & \$\\ \hline\end{array}$
b. Write a symbolic rule that expresses V in terms of x.
c. Determine the expected value of her home in 2016. (Simplify your answer)
d. In which year is the value of her home expected to reach $177,400?

An electric utility company determines the monthly bill by charging 85.4 cents per​ kilowatt-hour (kWh) used plus a base charge of ​$17.55 per month.​ Thus, the monthly charge is given by the function ${\displaystyle f\left(W\right)=0.854W+17.55}$ ​dollars, where W is the number of​ kilowatt-hours.
a. Find f (6000​) and explain what it means.
b. What is the monthly charge if 9500 kWh are​ used?
a. Select the correct choice below and fill in the answer box to complete your choice.
​(Simplify your answers. Type integers or​ decimals.)
A. ${\displaystyle f\left(6000\right)=}$ nothing.
The charge for 6000 kWh is $nothing.
B. ${\displaystyle f\left(6000\right)=}$ nothing.
The charge for nothing kWh is ​$6000.
b. The monthly charge is ​$nothing if 9500 kWh are used.
​ (Simplify your answer. Type an integer or a​ decimal.)

Solve the exponential equation. Express the solution set so that (a) solutions are in exact from and, if irrational, (b) Support your solutions by using a calculator.
${\displaystyle 3{\left(3\right)}^{x-3}+2=77}$
a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is ______.
B. The solution is the empty set.
b) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is _____.
B. The solution is the empty set.

Adding Decimals with Different signs. In January, the weight of a lion at a zoo changed by 1.8 pounds. In February, its weight changed by -2.3 pounds. What was the overall increase or decrease in the lion's weight?

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answer to three decimals.
${\displaystyle g\left(x\right)={\left(\frac{1}{3}\right)}^{x+1};g\left(\frac{1}{2}\right),g\left(\sqrt{2}\right),g(-3.5),g(-1.4)}$

You should answer this question WITHOUT using a CALCULATOR.
Write an equation ${\displaystyle \ln y=mx+b}$ form for the linear relationship shown in the table.
Reading a Book.
$\begin{array}{|cc|}\hline \text{t amount of time reading (hrs.)}& \text{P number of pages left to read}\\ 0& 420\\ 0.25& 405\\ 1.5& 330\\ 3& 240\\ 6& 60\\ \hline\end{array}$
DO NOT USE ANY SPACES between variables, constants, equals signs, and operation signs.
For example DO NOT enter ${\displaystyle y=2x+1}$ DO ENTER: ${\displaystyle y=2x+1}$
Put parentheses around constants that are fractions like this: ${\displaystyle y=\left(\frac{2}{3}\right)x-\left(\frac{1}{2}\right)}$
You may use decimals if the decimal values are exact. (You cannot use $0.33$ for ${\displaystyle \frac{1}{3}}$ because it is not exact, but you can use $0.25$ for ${\displaystyle \frac{1}{4}}$)
To write your equation, use the variables:
${\displaystyle t=}$ amount of time spent reading (hours)
${\displaystyle P\left(t\right)=}$ number of pages left to read.
The equation ${\displaystyle \ln y=mx+b}$ form is _________.

Solve each of the following equations for x. Determine the exact value for x and a decimal approximation correct to four decimals places.
1. ${\displaystyle 8x-15=0}$(Solving this equation will locate the x-intercept on the gtaph of ${\displaystyle y=8^x-15}$)
2. ${\displaystyle \log (x+6)=\log (3x-2)+2}$ (Solving this equation will determine the x-coordinate of the point of intersection of the functions ${\displaystyle y=\log (x+6)}$ and ${\displaystyle y=\log (3x-2)+2.)}$
3. ${\displaystyle 4400=2000{(1+\frac{0.045}{12})}^{12x}}$ (Solving this equation will determine the amount of time it takes an investment $2000 to grow to $4400 if interest is ${\displaystyle 4.5\mathrm{\%}}$ compounded monthly.)

Find the average rate of change of the function ${\displaystyle f\left(x\right)=5x}$ from ${\displaystyle x_1=0}$ to ${\displaystyle x_2=3}$.
The average rate of change is _______. (Simplify your answer.)

Grayson deposited $4,500 into a new account.
- The account earned annual simple interest.
- No withdrawals or deposits were made.
- At the end of 2 years, the balance of the account was $5,265.
What is the annual interest rate on this account?
${\displaystyle I=Prt}$, plug in numbers and solve equation in your calculator.
Convert ALL answer choices to decimals (percent divided by 100)
A. ${\displaystyle 4.7\mathrm{\%}}$
B. ${\displaystyle 5.2\mathrm{\%}}$
C. ${\displaystyle 7.3\mathrm{\%}}$
D. ${\displaystyle 8.5\mathrm{\%}}$

Find any holes, vertical asymptotes, and horizontal asymptotes for the following rational function.
${\displaystyle f\left(x\right)=\frac{2(x-3)(3x+5)}{3(3x+4)(x-3)}}$
The hole(s) is _____
The Vertical Asymptote(s) is _______
The Horizontal Asymptote is _______

A rectangular sandbox is surronded on all sides by a strip of grass that is 5 feet wide on all sides. The length of the sandbox is 4 feet more than the width, and the total area for the sandbox and the grass is 189 square feet. Find the dimensions of the sandbox.
The length of the sandbox is ___ feet and the width is ___ feet.
(Simplify your answers. Type integers or decimals rounded to the nearest hundredth as needed.)

In 2000, the life expectancy of females born in certain country was 79.9 years. In 2005, it was 82.1 years. Let E(t) represent the life expectancy and t the number of years after 2000. Assume that a constant rate of change exists for the model formed.
a) Find a linear function that fits the data.
b) Use the function of part (a) to predict the life expectancy of females in 2020.
a) Let E(t) represent the life expectancy and t the number of years after 2000. Write the linear function.
${\displaystyle E\left(t\right)=?}$
b) The predicted life expectancy of females in 2020 is ?

A woman's shoe that is size x in country A is size s(x) in country B, where ${\displaystyle s\left(x\right)=x-12}$. A woman's shoe that is size x in country ${\displaystyle p\left(x\right)=x-1.8}$. Find a function that will convert country A shoe size to country C shoe size.
The expression for the function is ___.
(Use integers or decimals for any numbers in the expression.)

A manufacturer of tools, selling rechargeable drills to a chain of home improvement stores, charges $2 more per drill than its manufacturing cost, m. The chain then sells each drill for ${\displaystyle 170\mathrm{\%}}$ of the price that it paid the manufacturer. Find a function P(m) for th price at the home improvement stores.
The function for the price at the home improvement stores is given by ${\displaystyle P\left(m\right)=?}$.

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answer to three decimals.
${\displaystyle h\left(x\right)=e^x;h\left(1\right),h\left(\pi \right),h(-3),h\left(\sqrt{2}\right)}$

Order the following decimals from least to greatest (separated by commas). 0.22, 0.67, 0.33, 0.6, 0.43

Compare the decimals.
8.86_________8.65

The number of full-time and part-time employees of an online retailer increased exponentially from 69 thousand in 2011 to 306.9 thousand in 2016.
a) Let ${\displaystyle t=0}$ correspond to 2011 and ${\displaystyle t=5}$ correspond to 2016. Then t is the number of years after 2011. Use the data points (0,69) and (5,306.9) to find the exponential growth rate and fit an exponential growth function ${\displaystyle P\left(t\right)=P_0{e}^{kt}}$ to the data, where P(t) is the number of employees, in thousands, t years after 2011.
b) Use the function found in part (a) to estimate the number of employees in 2017.
c) According to this model, when will there be one million employees?
a) P(t)=?

Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible; otherwise, use a calculator to approximate to two decimals.
${\displaystyle {\log }_8(x+5)-{\log }_8(x-2)=1}$