Question

A van is carrying a large amount of money in cash to deposit it in two ATM machincs on a hill station. The location of these machines is at the turning points of the path traced by the van, given by the equation h(x) = 2x³ − 18x² + 48x + 3, (x ≥ 0) where h(x) is the height of the hill (in 100 m) at any point.x.

1. Prove that the van is at the height of 300 m when it starts moving.   [1]
2. Find the location of the two ATM machines.   [2]
3. Calculate the difference between the heights of the location of the two ATM machines.   [1]
4. If the difference in the height of the location of the two ATM machines is greater than 1 km, then an extra armed security guard will be required. Based on the difference calculated in subpart (iii), determine if an extra armed guard will be required to protect the van.   [1]
5. Find the absolute maxima and absolute minima for h(x) in [0,4].    [1]

Answer 1

(i) When the van starts moving, its horizontal position x = 0.

Substitute x = 0 in the equation: 

h(0) = 2(0)³ − 18(0)² + 48(0) + 3

h(0) = 3

Since the height is in units of 100 m, the actual height is:

Height = 3 × 100 m  = 300m

(ii) The ATMs are at the turning points, where the derivative h′ (x) = 0

Differentiate h(x) with respect to x:

h′ (x) = 6x² − 36x + 48

Set h′(x) = 0

6x2 − 36x + 48 = 0   ...[Dividing by 6]

x² − 6x + 8 = 0

(x − 4)(x − 2) = 0    ...[Factorizing the quadratic equation]

x = 2 and x = 4

The locations of the two ATM machines are at x = 2 and x = 4

(iii) Difference b/w height 

4300 − 3500 = 800m

(iv) Extra guard needed if height difference > 1 km (1000 m)

calculated difference 800m

Since 800m < 1000m, an extra armed security guard will not be required to protect the van.

(v) To find absolute extrema in a closed interval, we check values at the endpoints and critical points within the interval

At x = 0 (endpoint): h(0) = 3

At x = 2 (critical point): h(2) = 43

At x = 4 (critical point/endpoint): h(4) = 35

Absolute Maxima = 43 (at x = 2)

Absolute Minima = 3 (at x = 0)