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. 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 . 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.
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 . Does this number fall short of, meet, or exceed the number of sales predicted by the formula?

Key Concepts

Mathematical Modeling

Mathematical modeling involves representing real-world situations with equations or functions to analyze and predict behaviors. It provides a structured framework for connecting theoretical concepts with practical applications, allowing for the estimation and interpretation of various phenomena over time.

Function Evaluation

Function evaluation is the process of substituting specific values into a mathematical expression or formula to compute the corresponding output. This technique is essential for understanding how a model behaves at precise points, making it possible to analyze and predict outcomes.

Logarithmic Functions

Logarithmic functions are mathematical operations that determine the exponent needed for a given base to produce a specific number. They model situations where growth or change occurs at a diminishing rate, which is common in natural and social phenomena.

Transcript

00:01 Here in this question, this whole question is about the cookies.

00:04 A new type of cookies is being launched.

00:06 The company predicts that it sales over the next 22 months will be approximated by the expression.

00:13 St is equal to 14 log piece 3, 2t plus 1, right? where t is the, you can say t is the number of months after the product has been introduced.

00:26 And st is simply the number of boxes in thousands.

00:30 Are being sold right so now in the first part we simply have to find out like how many boxes was sold after dapo on the market for one month so for one month that means you'll just plug in the value of t as one so simply just find out s one so s of one will be 14 log base 3 2 t plus 1 2 times 1 is 2 so 2 plus 1 that is 3 right now we all know that law base a times a is equal to 1 so what you will get simply 14 times 1 that is equal to 14 now in the question itself it was stated that s is the number of boxes in thousands so simply just multiply that by thousand so i can see that 14000 boxes of cookies was sold in the first month right now moving on to the second part the second part is asking you like how many boxes will be sold after they are in the market for four months so for four months so for four 4 months that means just plug in the value of 4.

01:33 So now we'll find out s of 4 that will be 14 log base 3 2 times 4 plus 1 that will be 9.

01:42 Now 14 log base 3 and 9 can be written as 3 squared right? so 14 times log with 3 3 square can be written as 14 times 2 which is equal to 28 and again the same thing s is the number of boxes in thousands.

02:00 So that means 28 ,000 of boxes will be sold after, like, after four months, the fee has been introduced, the total sale will be 38 ,000.

02:11 Now, coming on to the last part, we'll be saying that after 13 months, sale was approximated as 43 ,000.

02:19 Does this number for short of or exceed the number of sales pretty better? so after 13 months, that means we'll plug in the value of yes, 13 and we'll just find out, s of 13 by the formula...

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