Refer to the given figure. A road 8 km long is constructed along the chord PQ of a circular plot of radius 5 km. Two more roads are to be constructed from an external point T to the circle and tangential to it at P and Q. If expenses are Rs. 12000 per km for constructing the new roads TP and TQ, find the total cost of the roads to be constructed. How can the quality be improved and the cost of construction of roads be controlled?

Asked by Topperlearning User | 19th Feb, 2014, 09:12: AM

Expert Answer:

Since, OT is perpendicular bisector of PQ, therefore

PR = RQ… (1)

ButPQ = 8 km (given)

PR + RQ = 8

PR + PR= 8[using (1)]

PR = 4

RQ = PR = 4 km... (2) [using (1)]

In right triangle ORP, we have:

OP2 = OR2 + PR2

OR2 = OP2 - PR2

OR2 = 25 - 16 = 9

OR = 3 km

Since, TP is tangent to circle with centre O and OP is its radius, therefore,

OP TP

[ The tangent at any point of a circle is perpendicular to the radius through the point of contact]

OPT = 90o

In right triangle OPT, we have:

OT2 = PT2 + OP2

(TR + OR)2 = PT2 + 25

(TR + 3)2 = PT2 + 25... (4)

In right triangle PRT, we have:

PT2 = TR2 + PR2

PT2 = TR2 + 16... (5) [using (2)]

From(4) and (5), we have:

(TR + 3)2 = (TR2 + 16) + 25

TR2 + 9 + 6TR = TR2 + 41

6TR = 32

TR = ... (6)

Now, from (5) and (6), we get

We know tangents drawn from an external point to a circle are equal in length.

So, TP = TQ

Total length of the roads TP and TQ = km

Total cost of roads = = Rs. 1,60000

The contractors, concerned engineers and officers should be honest and persons of integrity and strong character. There should be no political interference and demands except honesty and fair control on the quality of construction. Then the country will shine.

Answered by  | 19th Feb, 2014, 11:12: AM