The heights of the `10^(th)` grade students of Manhattam Public School are measured . It is found that in Section A, the minimum height is `(x^(2) + 1)` feet while the maximum is (4x -1) feet. In section B, the minimum height is `(x^(2) + 1.5)` feet while the maximum is (4x - 0.5) feet. If all the students are considered , their range (i.e., difference between the maximum and minimum height) comes to 2.5 feet. What is the height of the tallest students (in feet) in the `10^(th)` grade , if both sections taken together ?

Answer

To solve the problem step by step, we will follow the logical flow of the information provided in the question.

### Step 1: Define the minimum and maximum heights for both sections
- For Section A:
- Minimum height = \( x^2 + 1 \) feet
- Maximum height = \( 4x - 1 \) feet

- For Section B:
- Minimum height = \( x^2 + 1.5 \) feet
- Maximum height = \( 4x - 0.5 \) feet

### Step 2: Determine the overall minimum and maximum heights
- The overall minimum height is the smaller of the two minimums:
\[ \text{Overall Minimum Height} = \min(x^2 + 1, x^2 + 1.5) = x^2 + 1 \]
(since \( 1 < 1.5 \))

- The overall maximum height is the larger of the two maximums:
\[ \text{Overall Maximum Height} = \max(4x - 1, 4x - 0.5) = 4x - 0.5 \]
(since \( -0.5 > -1 \))

### Step 3: Set up the equation for the range
- The range is defined as the difference between the maximum and minimum heights:
\[ \text{Range} = \text{Overall Maximum Height} - \text{Overall Minimum Height} = (4x - 0.5) - (x^2 + 1) \]

- According to the problem, this range is given as 2.5 feet:
\[ (4x - 0.5) - (x^2 + 1) = 2.5 \]

### Step 4: Simplify the equation
- Rearranging the equation:
\[ 4x - 0.5 - x^2 - 1 = 2.5 \]
\[ 4x - x^2 - 1.5 = 2.5 \]
\[ -x^2 + 4x - 1.5 - 2.5 = 0 \]
\[ -x^2 + 4x - 4 = 0 \]

### Step 5: Rearranging to standard quadratic form
- Multiplying through by -1 to make the leading coefficient positive:
\[ x^2 - 4x + 4 = 0 \]

### Step 6: Factor the quadratic equation
- This can be factored as:
\[ (x - 2)^2 = 0 \]

### Step 7: Solve for \( x \)
- Setting the factor equal to zero gives:
\[ x - 2 = 0 \implies x = 2 \]

### Step 8: Calculate the height of the tallest student
- The height of the tallest student is given by:
\[ \text{Tallest Height} = 4x - 0.5 \]
Substituting \( x = 2 \):
\[ \text{Tallest Height} = 4(2) - 0.5 = 8 - 0.5 = 7.5 \text{ feet} \]

### Final Answer
The height of the tallest student in the 10th grade, if both sections are taken together, is **7.5 feet**.

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