Domain and range of functions

[Damo17]

Can't figure out how to do this question. Any help would be appreciated.

The outline of an igloo can be modelled by the hybrid of two equations. The first part of the
igloo is modelled by ,where x is measured in metres.
a)If the curve passes through (0, 0) and (6, 6), find the value of a.
b)If this model applies for the first 10 metres, find the domain and range of this
function.
c)The remaining outline is modelled by . State its domain and range.

The entrance to the igloo is situated directly below the highest point of the igloo. The entrance is 2 m high at its peak and 4 m wide at its base. The entrance can be modelled by

d)Find the value of m.
e) Three children living in the igloo like to race each other
home and attempt to charge through the entrance
together. If the children are 1.6 m tall and their combined
width is 1.5 m, can they fit through the entrance at the
same time?

[/0]

a)




b)

If it applies for first 10m, then domain = [0, 10] (as we are assuming the igloo is above ground level)

Since the graph is basically a parabola which has been squashed by a square root sign, we can find the range by completing the square:



the maximum of the is y = 36 at x = 6 and the maximum of is y = 6 at x = 6.

To find the minimum, plug in x = 0 and x = 10:



Hence, range is [0, 6]. Sketching the graph and this will be easier to understand (when I do this question I think of the graph shape).

c) Solve to find the second intercept.

.

Therefore the domain is (10, 15].
Since is monotonically decreasing, plugging in x = 10 will get you the max, and x = 15 the min.




range =

d) Notice that .

i.e. the graph is the top half of a circle of radius 2, centred (6, 0). Hence, m = 2.

e) Take the children to be a rectangle of height 1.6m and width 1.5m.

To find the width allowable at a height of 1.6m, solve





As the distance , the children will easily fit through.

[Damo17]

^
Now I get it, thanks so much.