Hi guys, have got this problem to solve, but simply no idea how to do it... New here, but wonder if any of you clever maths people would mind giving it a go,
Much appreciated!
Some planners are designing a sports field. Their idea is to have a 200m running track, with the space in the middle to be used for other sporting events. The design has the running track composed of two parallel straight portions and two semicircular ends. The perimeter of the inside lane of the track must be 200 metres.
Length v, width 2t
a. If the length of one of the straight portions of the track is v and the radius of each of the semicircular ends is t, find an expression for the perimeter of the (inside lane of the) track and hence find a relationship between v and t
b. The rectangular area enclosed by the straight portions of the track and the diameters of the end semicircles is to be used for other sports. Thus, the designers want to make this rectangle as large as possible whilst keeping the perimeter of the track fixed at 200m
I. Find an expression for the area of this rectangle, firstly in terms of v and t, then in terms of t only
II. Find an expression for the area of this rectangle in terms of v only
III. Use differentiation to find the values of v and t which gives the area of this rectangle a stationary value, and verify that the stationary value is a (local) maximum of the area
Much appreciated!
Some planners are designing a sports field. Their idea is to have a 200m running track, with the space in the middle to be used for other sporting events. The design has the running track composed of two parallel straight portions and two semicircular ends. The perimeter of the inside lane of the track must be 200 metres.
Length v, width 2t
a. If the length of one of the straight portions of the track is v and the radius of each of the semicircular ends is t, find an expression for the perimeter of the (inside lane of the) track and hence find a relationship between v and t
b. The rectangular area enclosed by the straight portions of the track and the diameters of the end semicircles is to be used for other sports. Thus, the designers want to make this rectangle as large as possible whilst keeping the perimeter of the track fixed at 200m
I. Find an expression for the area of this rectangle, firstly in terms of v and t, then in terms of t only
II. Find an expression for the area of this rectangle in terms of v only
III. Use differentiation to find the values of v and t which gives the area of this rectangle a stationary value, and verify that the stationary value is a (local) maximum of the area