Q.65 A political party orders an arch for the entrance
to the ground in which the annual convention is being held. The profile
of the arch follows the equation
y = 2x − 0.1x2 where y is the height of the arch
in meters. The maximum possible height of the arch is
Concept Used
The given equation is a quadratic equation of the form:
y = ax2 + bx + c
- If a < 0, the parabola opens downward
- The maximum value occurs at the vertex
- The x-coordinate of the vertex is given by:
x = −b / (2a)
Step-by-Step Solution
Step 1: Identify coefficients
Rewriting the equation:
y = −0.1x2 + 2x
Here,
- a = −0.1
- b = 2
Step 2: Find x-coordinate of maximum height
x = −b / (2a)
x = −2 / [2(−0.1)] = 10
Step 3: Find maximum height
Substitute x = 10 in the equation:
y = 2(10) − 0.1(10)2
y = 20 − 10 = 10
Final Answer
Maximum possible height of the arch = 10 meters
Key Takeaway
When a quadratic equation has a negative coefficient of x2,
the maximum value occurs at the vertex. Using the vertex formula
provides a quick and accurate solution in exams.
