Help please Students in a seventh grade class were given a p

Help please:

Students in a seventh grade class were given a pre-algebra exam based on the material they learned that year. During the next two years the same students were retested once a month. The average score g can be approximated by the model g(x) = 87-16 log_10(t + 1), 0 lessthanorequalto t lessthanorequalto 24 Where t is the time in months. i. What was the average score on the original exam? ii. What was the average score after 6 months? iii. When did the average score drop below 70?

Solution

We have the average score in 24 monthly tests, g (t) = 87 - 16 log₁₀ ( t + 1) , 0 t 24. Therefore,

i. The average score on the original exam was g (t) + 16 log₁₀ ( t + 1) - 87.

ii. The average score g( 6) after 6 months will be obtained by substituting t = 6 in the expression for g (t)i.e. g( 6) = 87 - 16log₁₀ ( 6 + 1) = 87 - 16log₁₀ 7 = 87 - 16(0.84509804) = 87 - 13.52 = 73.47 or 73 (on rounding off )

iii. When the average score drops below 70, we have g (t) < 70 or,  87 - 16 log₁₀ ( t + 1) < 70 or, 87 - 70 < 16 log₁₀ ( t + 1) or, 16 log₁₀ ( t + 1) > 17 or log₁₀( t + 1)¹⁶ > 17 or ( t+ 1)¹⁶ > 10¹⁷ or, t + 1 > 10^17/16 so that t > 10^17/16 - 1 i.e t > 10.55 Thus, the average score falls below 70 when t 11 i.e. in the 11th month after the original exam or later.