A geologist you spoke with is concerned about the rate of land erosion around the base of a dam. Another geologist is studying the magma activity within the earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After they explain the concept and process to you, you decide to try and solve some problems.
1. Use synthetic division to determine if the first set of numbers are zeros of the given polynomial
a. -3, 2. f(x) = 3x^3 + 5x^2 - 6x + 18.
a. -4, 2. f(x) = 3x^3 +11x^2 - 2x + 8.
2. Given the polynomial f(x) = 2x^3 -5x^2-4x +3, find the solutions if the function is completed as
a) f(x)= 0
b) f(x +2)=0
d) f( 2x) =0
3. Today you visited the R&D (Research and development) development of a pharmaceutical manufacturer. The people in this lab have to calculate the recommended dosage of the products they are developing. Medicine is expressed as a function of the concentration in parts per million, with the potency of the dosage declining over a period of time. The function N(t) = ( 0.8t + 1000) / (5t +4 ), where t= > 15 gives the body concentration N(t), in parts per million , of a certain dosage of medication after time t, in hours. Find the horizontal asymptote of the graph and complete the following statement: N(t) approaches ? as t approaches infinity. Explain the meaning of the answer to a) in terms of the medical dosage application.
4. A retailer you spoke with in New York City's fashion district imports haute couture from European designers . One of the accommodations which must be considered when importing fashion from other countries is the difference in the size charts. A function that will convert dress sizes in the United States to those in Italy is g(x) = 2(x + 12). Find the dress sizes in Italy that correspond to sizes 6, 10, and 14 in the United States. Find a formula for the inverse of the function . Use the inverse function to the dress sizes in the United States that correspond to 36, 44 , and 60 in Italy.
5. Last week's profits from a dry cleaners was $2000. suppose the $2000 is invested at interest rate k, compounded continuously, and grows to 2983.65 in 5 years.
a. what is the interest rate?, b. find the exponential growth function, c. what will the balance be after 10 years ?, c. when will the $ 2,000 double?
There are several types of problems here explaining synthetic division, inverse functions, horizontal asymptotes (and what it means in context), and interest rate. This solution shows all work and answers attached in a Word document.