Class 12 Applications Of Derivatives · Case Study4 Markscase_based_application
Class 12 Maths · Applications Of Derivatives · Case Study #32
A served volleyball has modeled height h(t)=−7t²/2+13t/2+1 metres at time t seconds after service. Use the polynomial for t≥0 mathematically; the physical flight ends at its first positive zero t=2 seconds.
Part (1) 2 Marks
Is h(t) continuous? Justify.
Answer: Yes, throughout its modeled time domain.
- h(t) is a polynomial in t, being a sum of constant multiples of powers of t.
- Every polynomial is continuous on R, so its restriction to t≥0 and to the physical flight [0,2] is continuous.
- Thus the modeled height is continuous throughout the flight.
Part (2) 2 Marks
Find the time at which the height is maximum.
Answer: 13/14 seconds
- Differentiate: h′(t)=−7t+13/2.
- Setting h′=0 gives t=13/14; h″=−7<0 and this point lies in [0,2].
- The global maximum occurs at t=13/14 seconds because the concave quadratic increases before and decreases after that instant.