2019 NJC P1 Q10

A construction company, which has just won a contract to build foundations at two different sites I and II, wants to buy a new borewell drilling machine. Two brands, MTZ and STK, are considered and tested. To test a machine, it is used to drill a hole for ten hours.

The following date were recorded.

MTZ: The depth drilled is \(4.5\)m in the first hour, and for each subsequent hour, the depth drilled decreases by a positive constant \(r%\).

STK: The depth drilled is \(4.7\)m in the first hour, and for each subsequent hour, the depth drilled decreases by \(0.1\)m.

1. Show that the total depth drilled by an MTZ borewell drilling machine is
   \(\frac{450}{r}\left[ 1-{{\left( 1-\frac{r}{100} \right)}^{10}} \right]m\).[2]
2. Find the value of \(r\) if the total depth drilled by both machines are the same.[2]

It is given that the machines perform the same way as recorded at the test site.

3. Assuming \(r=1\), to maximise the efficiency of the machine, the company will stop drilling if the depth to be drilled in the next hour is less than \(4.0\)m at this site. How many complete hours should the machine be operated?[2]

Before drilling starts at Site II, the engine of MTZ borewell drilling machine breaks down.

4. To save costs, the company considers to buy a new engine from an alternative manufacturer. The manufacturer claims that with the new engine, the MTZ borewell drilling machine will be able to drill a hole to a theoretical maximum depth of at least \(360\)m if it is operated continuously. Find the range of the values of \(r\) for which the alternative manufacturer’s claim is false.[3]
5. The company changes its mind and decides to buy a STK borewell drilling machine to drill the hole at Site II instead. Suppose that the machine performs the same way at Site II as at the test site, explain clearly whether the STK borewell drilling machine will be able to drill a hole of depth \(150\)m if it is operated continuously.[3]