(a) When the lake is at it's largest, what are the coordinates of it's vertices? Explain...

 

Transcribed Image Text:

(a) When the lake is at it's largest, what are the coordinates of it's vertices? Explain your reasoning. (b) Find where the farmer should build his new fences, such that they will enclose the lake at any time of year, and will minimise the length of fencing needed (so he wants to make the shortest possible straight-line fences). Find equations which describe these fences in this cartesian coordinate system, and the coordinates of the fence-posts to which they will attach. (c) What is the total length of fencing required for these new fences? (d) Given that the fences need fence posts to support them, and the distance between fence posts cannot be any more than 10m, what is the minimum number of new fence posts that will need to be added? Remember, the tree and existing fence posts are already there, so do not need to be counted. Below is a top-down view of the field in which the lake sits, marked out with a set of Cartesian axes measured in units of metres. There is a fence that runs parallel to the y-axes, with a post every 10m described by the equation x = 35, and a tree at coordinates (5,20). The centre of the lake has coordinates (25, 25), and because the lake is formed at the intersection of four hills, it is approximately square, as shown in the diagram below. The size of the lake in this picture changes depending on how full it is, and the side length of the square shape of the lake, s, can be related to the depth of the water in the lake d by the equation s = d 10. The farmer wants to fence off the lake from the rest of the field by putting up two new fences, each making a straight line from the tree to one of the existing fence posts. In this question, you will design where these new fences should go. 25 (5,20) S x = 35 (35,40) (35, 30) (25,25) S (35,20) (35, 10) (35,0) 25 25 x (a) When the lake is at it's largest, what are the coordinates of it's vertices? Explain your reasoning. (b) Find where the farmer should build his new fences, such that they will enclose the lake at any time of year, and will minimise the length of fencing needed (so he wants to make the shortest possible straight-line fences). Find equations which describe these fences in this cartesian coordinate system, and the coordinates of the fence-posts to which they will attach. (c) What is the total length of fencing required for these new fences? (d) Given that the fences need fence posts to support them, and the distance between fence posts cannot be any more than 10m, what is the minimum number of new fence posts that will need to be added? Remember, the tree and existing fence posts are already there, so do not need to be counted. Below is a top-down view of the field in which the lake sits, marked out with a set of Cartesian axes measured in units of metres. There is a fence that runs parallel to the y-axes, with a post every 10m described by the equation x = 35, and a tree at coordinates (5,20). The centre of the lake has coordinates (25, 25), and because the lake is formed at the intersection of four hills, it is approximately square, as shown in the diagram below. The size of the lake in this picture changes depending on how full it is, and the side length of the square shape of the lake, s, can be related to the depth of the water in the lake d by the equation s = d 10. The farmer wants to fence off the lake from the rest of the field by putting up two new fences, each making a straight line from the tree to one of the existing fence posts. In this question, you will design where these new fences should go. 25 (5,20) S x = 35 (35,40) (35, 30) (25,25) S (35,20) (35, 10) (35,0) 25 25 x