Respuesta :
Answer:
[tex]\mathbf{S(t)=200(\frac{105}{100})^{x}}[/tex]
[tex]\mathbf{A(t)=40(\frac{98}{100})^{x}}[/tex]
[tex]\mathbf{E(t)=S(t) \cdot A(t)=200(\frac{105}{100})^{x} \cdot 40(\frac{98}{100})^{x}=8000(\frac{10290}{10000})^{x}}[/tex]
Step-by-step explanation:
The predicted number of students over time, S(t)
Rate of increment is 5% per year.
A function 'S(t)' which gives the number of students in school after 't' years.
S(0) means the initial year when the number of students is 200.
S(0) = 200
S(1) means the number of students in school after one year when the number increased by 5% than previous year which is 200.
S(1) = 200 + 5% of 200 = [tex]200+\frac{5}{100}\time200[/tex] = [tex]200(1+\frac{5}{100})[/tex] = [tex]200(\frac{105}{100})[/tex]
S(2) means the number of students in school after two year when the number increased by 5% than previous year which is S(1)
S(2) = S(1) + 5% of S(1) = [tex]\textrm{S}(1)(\frac{105}{100})[/tex] = [tex]200(\frac{105}{100})(\frac{105}{100})[/tex] = [tex]200(\frac{105}{100})^{2}[/tex]
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Similarly [tex]\mathbf{S(x)=200(\frac{105}{100})^{x}}[/tex]
The predicted amount spent per student over time, A(t)
Rate of decrements is 2% per year.
A function 'A(t)' which gives the amount spend on each student in school after 't' years.
A(0) means the initial year when the number of students is 40.
A(0) = 40
A(1) means the amount spend on each student in school after one year when the amount decreased by 2% than previous year which is 40.
A(1) = 40 + 2% of 40 = [tex]40-\frac{2}{100}\time40[/tex] = [tex]40(1-\frac{2}{100})[/tex] = [tex]40(\frac{98}{100})[/tex]
A(2) means the amount spend on each student in school after two year when the amount decreased by 2% than previous year which is A(1)
A(2) = A(1) + 2% of A(1) = [tex]\textrm{A}(1)(\frac{98}{100})[/tex] = [tex]40(\frac{98}{100})(\frac{98}{100})[/tex] = [tex]40(\frac{98}{100})^{2}[/tex]
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Similarly [tex]\mathbf{A(x)=40(\frac{98}{100})^{x}}[/tex]
The predicted total expense for supplies each year over time, E(t)
Total expense = (number of students) × (amount spend on each student)
E(t) = S(t) × A(t)
[tex]\mathbf{E(t)=S(t) \cdot A(t)=200(\frac{105}{100})^{x} \cdot 40(\frac{98}{100})^{x}=8000(\frac{10290}{10000})^{x}}[/tex]
[tex]\mathbf{E(t)=8000(\frac{10290}{10000})^{x}}[/tex]
(NOTE : The value of x in all the above equation is between zero(0) to ten(10).)
