To solve the two parts of your question, we will first evaluate the formula for the year 2010 and then find when the predicted percentage reaches 58%.
Part 1: Calculate the percentage for 2010 using the formula
The formula given is:
\[ p = \frac{4x}{5} + 26 \]
where \(x\) is the number of years after 1980.
For the year 2010:
- \(x = 2010 - 1980 = 30\)
Now substitute \(x\) into the formula:
\[ p = \frac{4(30)}{5} + 26 \]
Calculating this step-by-step:
- \(4(30) = 120\)
- \(\frac{120}{5} = 24\)
- \(p = 24 + 26 = 50\)
According to the formula, the percentage of college freshmen with an average grade of A in 2010 is 50%.
Comparison with the bar graph:
From the bar graph, the percentage for the year 2010 is 46%.
To determine whether the model underestimates or overestimates, we compare:
- Formula result: 50%
- Actual value from graph: 46%
Since 50% (model) is greater than 46% (actual), this means the model overestimates the actual value.
The overestimate is:
\[ 50 - 46 = 4% \]
Thus, the answer to Part 1 is: B. The model overestimates the actual value by 4%.
Part 2: Project when 58% of college freshmen will have an average grade of A in high school.
To find when \(p = 58\), we set up the equation:
\[ 58 = \frac{4x}{5} + 26 \]
To solve for \(x\), first isolate the term with \(x\):
- Subtract 26 from both sides:
\[ 58 - 26 = \frac{4x}{5} \] \[ 32 = \frac{4x}{5} \]
- Multiply both sides by 5 to eliminate the fraction:
\[ 32 \times 5 = 4x \] \[ 160 = 4x \]
- Divide both sides by 4:
\[ x = \frac{160}{4} = 40 \]
Now, convert \(x\) back to the actual year by adding it to 1980:
\[ \text{Year} = 1980 + x = 1980 + 40 = 2020 \]
Thus, the predicted year when 58% of college freshmen will have an average grade of A in high school is:
2020.
Summary Answer:
- Part 1: B. The model overestimates the actual value by 4%.
- Part 2: The year predicted is 2020.