- #1
Saracen Rue
- 150
- 10
Homework Statement
Sally makes a window frame which can be modeled by the composite function ##f\left(x\right)=\left[-8e^{\frac{x}{30}}+160\left\{0\le x\le a\right\},\ -8e^{\frac{-x}{30}}+160\left\{-a\le x<0\right\}\right]##, where ##a## is the x-intercept. A second composite function, ##g\left(x\right)=\left[50\log _n\left(\frac{x}{2}+1\right)\left\{0\le x\le b\right\},50\log _n\left(1-\frac{x}{2}\right)\left\{-b\le x\le 0\right\}\right]## (where ##b## is the x-coordinate of the point of intersection), models the bounds of the region which is to be filled with stained glass as shown:
Given that all units are in centimeters and that ##1<n≤10##;
(a) Find the exact value of the constant, ##a##
(b) Express the area of the region which is to be filled with stained glass as a function of ##n## and in terms of ##b##.
(c) State the domain and range of the ##A(n)##, correct to 3 decimal places
(d) The cost of stained glass is ##$450## per square meter and the cost of regular glass is ##$200## per square meter. If Sally can only spend ##$800##;
i) Find the maximum area of stained glass Sally could use correct to 4 decimal places
ii) Determine the value of ##n## for this area, correct to 1 decimal place.
ii) Determine the value of ##n## for this area, correct to 1 decimal place.
Homework Equations
Integrating between two points on a graph gives the area under the graph between said points
The Attempt at a Solution
I was able to do parts (a) and (b) without much trouble, but I'm rather stuck at part (c). I understand that the restriction on ##n## which was given as part of the question acts as the domain of ##A(n)##, but what I don't understand is how exactly the range part works. The lowest possible area occurs at ##n=10## and is inclusive, which is fine. The largest area is confusing me though - I tried making ##n=1## to find the area at the point, and then specifying that that point is not inclusive (which is how I've always done this sort of question). However, when I do this I get not-defined as an answer. Can anybody help please?