A company plans to introduce a new type of cookie to the...

A company plans to introduce a new type of cookie to the market. The company predicts that its sales over the next 24 months can be approximated by $$S(t)=14 \log _{3}(2 t+1)$$ where $t$ is the number of months after the product is introduced, and $S(t)$ is in thousands of boxes of cookies. (IMAGE CAN NOT COPY) a) How many boxes of cookies were sold after they were on the market for 1 month? b) How many boxes were sold after they were on the market for 4 months? c) After 13 months, sales were approximately 43,000 dollar Does this number fall short of, meet, or exceed the number of sales predicted by the formula?

A company plans to introduce a new type of cookie to the market. The company predicts that its sales over the next 24 months can be approximated by
$$S(t)=14 \log _{3}(2 t+1)$$
where $t$ is the number of months after the product is introduced, and $S(t)$ is in thousands of boxes of cookies.
(IMAGE CAN NOT COPY)
a) How many boxes of cookies were sold after they were on the market for 1 month?
b) How many boxes were sold after they were on the market for 4 months?
c) After 13 months, sales were approximately 43,000 dollar Does this number fall short of, meet,
or exceed the number of sales predicted by the formula?

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical functions that use a logarithm to solve for the exponent in an exponential equation. They are particularly useful for modeling processes that involve rapid initial growth which then slows over time. Understanding their properties, such as the base of the logarithm and how they behave graphically, is key when they are used in various mathematical models.

Mathematical Modeling

Mathematical modeling involves creating equations or functions to represent real-world situations. In this context, a logarithmic model is applied to predict sales over time, demonstrating how mathematical techniques can be used to forecast outcomes and analyze trends. This process requires translating practical scenarios into formal mathematical expressions.

Function Evaluation

Function evaluation is the process of computing the output of a function for a given input. It requires substituting specific values into the function’s formula and performing the necessary calculations to obtain numerical results. This concept is fundamental in many areas of mathematics, especially in applied fields where functions represent real-world quantities.

Unit and Scale Analysis

Unit and scale analysis involves careful attention to the units of measurement and the scales used in a model. It ensures that computed results are meaningful and correctly interpreted, particularly when converting between different measurement units or when scaling values (such as thousands or millions) to match real-world observations.

Comparison of Predictions and Empirical Data

Comparing predictions from a mathematical model to actual observed data is a critical step in validating the model. This process involves calculating expected outcomes using the model and then evaluating whether real-world measurements meet, exceed, or fall short of these predictions. Such comparisons help in assessing the accuracy and applicability of the model in practical scenarios.

Transcript

00:01 This word problem, we are given an equation s of t is equal to 14 times log base 3 of 2t plus 1.

00:07 And this represents the relationship between a company sales of cookies over 24 months, where t stands for the number of months.

00:17 So for part a, we are looking, so s of t is in thousands of box of cookies.

00:26 S of t, a thousand box of cookies.

00:33 So it's saying.

00:33 Thousand times.

00:35 So okay, part a, we are asked how many boxes were sold after one month.

00:45 So t equals one.

00:47 And we simply use, we substitute this value into the equation over here and we get 14, log three, two t, two times one is two plus one is three.

01:02 So we get three and we get.

01:06 So we know that for a number to, in a logarithm, for a base to have an argument of the same number, only a number that is raised to the one power is able to do so.

01:20 So this logarithm has to be equal to one.

01:23 So therefore, we have s of t equals 14.

01:29 But remember, we still need a thousand times 14 because each unit of st represents 1 ,000 cookies.

01:37 So 1 ,000 times 14 equals 14 ,000 cookies.

01:44 And that is the answer to part a.

01:48 So for part b, they're asking us how many cookies cookies will have sold after four months.

01:55 So t equals 4...