00:01
In the given question the following figure is given to us and we are told that a power line is to be installed across a river that is one meter one mile wide to a town that is five miles downstream.
00:14
So this is the figure and we can see that the width of this river is given as one miles, right? it is given to be one mile the town is located five miles away from this point it is five miles away and now we are told that the power line is to be installed across this river like this so this distance to the point across the river that the power line would be installed is at a distance of x from this point right so now we are told that it cost 7 ,500 per mile to lay the cab to lay the cable under water so let's write this down to lay what to lay cable to lay the cable under water to lay the cable under water underwater it cost 7 ,500 per mile it cost 7 ,500 dollars per mile and we are told that it cost $6 ,000 per mile to lay it over land so to lay the cable to lay the cable over land over land it cost $6 ,000 per mile.
01:56
And now we are told to determine how the cable should be installed if the limit is 35 ,000 that is allocated for this project.
02:07
So what we could do over here is we can see that from this figure what we have is a right angled triangle over here and in this right angled triangle, the distance, the distance the base of this triangle is having a length of x miles.
02:26
The height is one mile and this is how we would lay the cable, right? lay the cable underwater.
02:35
So let's find this distance which is the hypotenuse of this triangle, right? so using the pythagorean theorem, using the pythagorean theorem, pythagorean theorem, pythagorean theorem, we can write that in a triangle, right angle to triangle with length a, b and c, where c is the length of the i portnus, we can write c is equal to the square root of a squared plus b square.
03:12
So this is what the pythagorean theorem gives us.
03:16
So now we could use this theorem over here and we could write the length of the hypotenuse over here would then be equal to the square root of x square plus one square which is x the square root of x square plus one.
03:32
So now that we know the distance that we need to lay the cable under water is x square plus one.
03:39
What we could do now is we can write if for one mile at a 7 ,500 per mile for laying cable for laying cable under water under water for the square root of x square plus one miles is equal to 7500 times the square root of x square plus one miles is equal to 7 ,500 times the square root of x square plus one and similarly now we should know that overland from this point that is the point at which this underwater cable reaches the land to the point where the power should be delivered to the town the distance to be covered now is 5 minus x miles right since x miles when we take the cable from the station to the other side to the other side of the river the square root of x square plus one miles is the distance that is the distance that we have laid cable underwater and this this has covered x miles on land so now the only thing that we need to the only distance that we need to lay cable to reach the town is 5 minus x miles so this means we can write for laying cable for laying cable, laying cable over land, over land for 5 minus x miles, 5 minus x miles.
05:24
The cost would be the cost is equal to 6 ,000 times 5 minus x.
05:36
Now according to the rule or the condition that is given in the question we are told that the allocated amount for this project is 35 ,000 right so we can write 7 ,500 7 ,500 times the square root of x square plus 1 plus 6 ,000 times 5 minus x should give us a total of 35 ,000.
06:08
So now we should just solve this to find the value of x...