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.
What is the number of bactenia ater 15 minutes? (round to the nearest whole number of bacteria.)

The sum is 110.Find two positive real numbers whose product is a maximum.

the range of this quadratic function
${\displaystyle y=-x^2+6x-2}$

How deep is the water in the trough if a water trough is in the shape of a rectangular prism that is 12 feet long by 3 feet wide and has 9.4 cubic feet of water?

An object is thrown upward from the ground with an initial velocity of ${\displaystyle 32\frac{ft}{s}}$. What is the maximum height the object obtains using the formula ${\displaystyle s=-16t^2+32t}$, where s = distance above the ground in feet, and t= time in seconds?

Using the concept of exponential growth and exponential decay. Solve the given problem.
1. A small locality has a population of 52,365 in 2012. If its population increases 3% every 2 years,
a. Derive a function P that determines the population t years after 2012.
b. What is the expected population in 2020?

Give an example of an exponential function in the form ${\displaystyle y=a\times b^x}$ that is neither an exponential growth function nor an exponential decay function. Explain your reasoning.

Which characteristic of an exponential decay function does not belong with the other three? Explain your reasoning. base of 0.8 decay factor of 0.8 decay rate of 20% 80% decrease

What is the asymptote of the graph of an exponential growth or exponential decay function? Explain your reasoning.

In 2011, the Population of China and India were approximately 1.34 and 1.19 billion people, respectively. However due to central control the annual population growth rate of China was 0.4% while the population of India was growing by 1.37% each year. if these growth rates remain constant. when will the population of India exceed that of China?

Factoring ${\displaystyle 4x^2-2xy-4x+3y-3}$
Why isn't it working?

Let p,q, and r be constants. One solution to the equation
${\displaystyle (x-p)(x-q)=(r-p)(r-q)}$
is ${\displaystyle x=r}$. Find the other soultion in terms of p,q, and r.

Determine the real number A such that the following equation has a real answer for C for every real ${\displaystyle B:A^2+B^2+C^2+2ABC=1}$

Proving that ${\displaystyle x^2-2xy+6y^2-12x+2y+41\ge 0}$ where x, ${\displaystyle y\in \mathbb{R}}$

An inequality constraint problem
${\displaystyle min\frac{1}{2}{(x_1-1)}^2+\frac{1}{2}{(x_2-2)}^2}$
s.t. ${\displaystyle x_1-x_2=1\text{and}{x}_1+x_2\le 2}$

quadratics by factoring
2z(5z-2)=-5z+2

How do we know that the quadratic ${\displaystyle 3y^2-y-12}$ has real root?
(a) Notice the quadratic cannot be factored into the product of two binomials with integer cofficients. Does this mean that the quadratic does not have any real roots?
(b) If the answer to part (a) is "no", then explain how we know that the quadratic does have real roots.
(c) Suppose the quadratic has roots ${\displaystyle y=r\text{and}y=s}$. Find a quadratic with roots ${\displaystyle r+2\text{and}s+2}$.

Sign of ${\displaystyle P\left(\Delta \right)}$ with ${\displaystyle P\left(x\right)=ax²+bx+c}$ and ${\displaystyle \Delta =b^2-4ac}$

How does the study of abstract algebraic structures help you understand "algebra" in general?(Here by "algebra" in general, I mean the equations and things you've worked with in previous algebra courses in high school and college.)

Determine whether the ordered pair is a solution to the given system of linear equations. (-1,7) 4x+y=3 2x-3=y