Master Algebra 1 Word Problems with our Comprehensive Practice

Exponential Growth and Decay
Exponential growth and decay problems follow the model given by the equation ${\displaystyle A\left(t\right)=P{e}^{rt}}$
-The model is a function of time t
-A(t) is the amount we have ater time t
-PIs the initial amount, because for t=0, notice how ${\displaystyle A\left(0\right)=P{e}^{0\times t}=P{e}^0=P}$
-Tis the growth or decay rate. It is positive for growth and negative for decay
Growth and decay problems can deal with money (interest compounded continuously), bacteria growth, radioactive decay. population growth etc.
So A(t) can represent any of these depending on the problem.
Practice
The growth of a certain bactenia population can be modeled by the function
${\displaystyle A\left(t\right)=900e^{0.0534}}$
where A(t) is the number of bacteria and t represents the time in minutes.
How long will t take for the number of bacteria to double? (your answer must be accurate to at least 3 decimal places.)

The half-life of a radioactive kind of xenon is 9 hours. If you start with 32 grams of it, how much will be left after 18 hours?

Solve problems involving antiditerentiation
Solve situational problems involving exponential growth and decay
1. The rate of decay of radium is said to be proportional to the amount of radium present. If the half-life of radium is 1690 years and there are 200 grams on hand now, how much radium will be present in 845 years?

Using the concept of exponential growth and exponential decay, solve the given problem. Show complete and systematic solutions.
1. An unknown radioactive element decreases 12% of its amount every 5 days. What is exponential function? that describes the amount left after t days? If there are 300g of the substance at present, how much is left after 30 days? (round off the value up to two decimal places)

Tell whether the function represents exponential growth or exponential decay. Then graph the function.
${\displaystyle y={\left(1.8\right)}^x}$

Rewrite the function to determine whether it represents exponential growth or exponential decay.
${\displaystyle y=2{\left(1.06\right)}^{9t}}$

Solve the following problems involving exponential growth and decay.
The half-life of carbon-14 1s approximately 6000 years. How much of 800 g of this substance will remain after 30,000 years?

Why do I get incorrect solutions ${\displaystyle x=0}$ and ${\displaystyle x=2}$ for ${\displaystyle x+1+\sqrt{4x+1}=0?}$

Solving quadratic matrix equation for X
$X^2=\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]$
where ${\displaystyle a\in \frac{\mathbb{R}}{0}}$. Solve for matrix X.

Solve fractional inequalities that have quadratics
${\displaystyle \frac{2x^2+3x-2}{2x^2-3x-2}>0}$
How do you find the range of solutions? Can you just bring the denominator onto the right hand side or does that lose solutions?

Exponential Growth and Decay
Exponential growth and decay problems follow the model given by the equation ${\displaystyle A\left(t\right)=P{e}^{rt}}$
-The model is a function of time t
-A(t) is the amount we have ater time t
-PIs the initial amount, because for t=0, notice how ${\displaystyle A\left(0\right)=P{e}^{0\times t}=P{e}^0=P}$
-Tis the growth or decay rate. It is positive for growth and negative for decay
Growth and decay problems can deal with money (interest compounded continuously), bacteria growth, radioactive decay. population growth etc.
So A(t) can represent any of these depending on the problem.
Practice
The growth of a certain bactenia population can be modeled by the function
${\displaystyle A\left(t\right)=900e^{0.0534}}$
where A(t) is the number of bacteria and t represents the time in minutes.
What is the initial number of bacteria? (round to the nearest whole number of bacteria.)

The complex zeros of f 1x2 = x4 + 1 For the function f1x2 = x4 + 1:
(a) Factor f into the product of two irreducible quadratics. (Hint: Complete the square by adding and
subtracting 2x2
.2
(b) Find the zeros of f by finding the zeros of each irreducible quadratic.

Find all values of a and b that will make
${\displaystyle (ax+5)2=16x2+bx+25}$ true for all x.

A certain radioactive substance has a half-life of 12 days. This means that every 12 days, half of the original amount of the substance decays. If there are 128 milligrams of the radioactive substance today, how many milligrams will be left after 48 days?

Integer between the 's p and ${\displaystyle p+2}$ is a perfect square. Explain why there exists an integer n such that
${\displaystyle 4n^2-1=p}$
a) Is the integer between p and ${\displaystyle p+2}$ odd or even?
b) Assume additionally that the integer between the 's p and ${\displaystyle p+2}$ is a perfect square.
c) By considering ${\displaystyle (2n-1)(2n+1)}$, find the only possible value of p.

Are there any two real nonzero numbers a and b such that ${\displaystyle a^2+ab+b^2=0}$ Why or why not?

This scenrio has do around QUADRATICS problems.
You have been asked to bake the birthday cake of a little girl who is about to turn three. When looking up suggestions online on how to bake this cake, you found that most everyone suggests baking the cake in smaller circular cake pans that have a diameter of 6 inches. That’s the best way to get the height for the unicorn head. A normal circular cake pan (and the only kind you have) has a diameter of 9 inches. You need to buy cake pans. As an experienced baker, you know that the recipe you are planning on using usually fills two regular (9-inch) cake pans, with a little room to spare. How many 6-inch pans do you need to buy? (We’re going to assume that all cake pans are the same height.)

Does every quadratic polynomial defined in real numbers always define a function?
For example, let a,b,c ${\displaystyle \in \mathbb{R}}$ and define
${\displaystyle P\left(x\right)=a{x}^2+bx+c}$