# Manipulating quadratic and exponential expressions

• MHB
• mdp448
In summary, the function L(t)=5.3 x 1.025^t shows the approximate percent literacy rate in India t years after 1900 and the equivalent function L(t)=5.3 x 3^t/44.5 represents the approximate number of years it took for the literacy rate to triple. The answer is obtained by setting 3 as the constant in the equation and calculating the value of t using logarithms. Equation (a) provides the correct approximation for the number of years.
mdp448
I am having so much trouble figuring this out, I would really appreciate some help.

The question is:
The following function, L, gives the approximate percent literacy rate in India t years after 1900.

L(t)=5.3 x 1.025^t

Which of the following equivalent functions shows, as a constant or coefficient, the approximate number of years it took for the literacy rate to triple?

(a) L(t)=5.3 x 3^t/44.5
(b) L(t)=5.3 x 1.077^t/3
(c) L(t)=5.3 x 1.008^3t
(d) L(t)=3 x 1.025^t+23

Thanks so much. I know what the answer is, but I just have no idea why it is the answer. I just want to understand :(

initial literacy percentage is $L(0) = 5.3 \cdot (1.025)^0 = 5.3$

triple literacy percentage is $3 \cdot 5.3$ ...

$3 \cdot 5.3 = 5.3 \cdot (1.025)^t$

$3 = 1.025^t$

$\log(3) = t\log(1.025) \implies t = \dfrac{\log(3)}{\log(1.025)} \approx 44.5 \text{ years}$

... now look at equation (a)

## 1. What is a quadratic expression?

A quadratic expression is an algebraic expression that contains a variable raised to the second power (x^2) and may also include a linear term (ax) and a constant term (b). It can be written in the form of ax^2 + bx + c, where a, b, and c are constants.

## 2. How do you manipulate a quadratic expression?

To manipulate a quadratic expression, you can use algebraic techniques such as factoring, completing the square, or using the quadratic formula. These methods can help you simplify or solve the expression for a given variable.

## 3. What is an exponential expression?

An exponential expression is an algebraic expression that contains a variable in the exponent. It can be written in the form of a^x, where a is the base and x is the exponent. It represents repeated multiplication of the base a by itself x times.

## 4. How do you manipulate an exponential expression?

To manipulate an exponential expression, you can use properties of exponents such as the product rule, quotient rule, and power rule. You can also use logarithms to solve for the variable in the exponent.

## 5. How are quadratic and exponential expressions related?

Quadratic and exponential expressions are related in that a quadratic expression can be rewritten as an exponential expression with a base of e (Euler's number) raised to the power of the quadratic expression. This relationship is known as the exponential form of a quadratic expression.

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