Interpret S(t) = 260 log(At + 10)-220

Question

A student has spent the last few years learning Spanish at a special language Academy. The number of words in Spanish,
S(t), that he knows is given below where A is a positive constant and t is the number of years after he started at the Academy.

S(t) = 260 log(At + 10)−220

(a) How many words in Spanish did the student know when he started the Academy?
0 words
260 words
40 words
480 words
220 words

For this I said 220 but I am not sure. Since the midline has a minus sign in front does that mean its a negative midline. Wouldn't that mean they would start at a negative 220 words or something.

(b) At the end of the first year at the Academy, the student knew 320 words in Spanish. Find the value of A to the nearest tenth.
119.4
109.4
1.9
17.0
7.0

I know how to do this up until a certain point. I equated the equation to 320 and then I subtracted 220 and divided by 260 but that's as far as I can get. I don't know how to do the rest. please help me.

Answer 1

S(t) = 260 log(At + 10) - 220

 

(a) Substitute t = 0 in the equation above and get

 

S(0) = 260 log(10) - 220 = 260 - 220 = 40

 

(b)

 

t = 1

S(1) = 320

 

320 = 260 log[A(1) + 10] - 220

 

540 = 260 log(A  + 10) ⇒ 2.0769 = log(A + 10)

 

A + 10 = 10^2.0769 ≅ 119.37

 

A = 119.37 - 10 ≅ 109.4 

 

Answer 2

S(t) = 260 log(At+10) - 220   where log is log base 10

 

a) At t=0 (when the student started):

 

    S(0) = 260 log(A*0+10) - 220

    S(0) = 260 log(10) - 220

 

    Now log(10) = 1

 

    S(0) = 260*1 - 220

    S(0) = 40 words

 

b)  When t=1 (end of first year):

    S(1) = 260 log(A*1+10) - 220

 

    Since S(1) = 320:

 

    320 = 260 log(A+10) - 220

    540 = 260 log(A+10)

    540/260 = log (A+10)

    10^540/260 = A + 10

 

Can you finish it from here? Use your calculator.

 

Answer 3

(a). S(0) = 260log10 - 220  = 260(1) - 220

 

                                       = 40 words

 

(b). 320 = 260log(A+10) - 220

 

      540 = 260log(A+10)

 

      2.0769 = log(A+10)

 

      A+10 = 10^2.0769 = 119.37

 

                           A = 109.37 ≈109.4