LINKING concepts... For Individual or Group Explorations Designing a Race Track An architect is designing a race track that will be 60 feel wide and have semicircular ends, as shown in the accompanying figure. The length of the track is to he one mile. measured on the inside edge of the track. a) Find the exact dimensions for x and y in the figure that will maximize the area of the rectangular center section. b) Round the answers to part (a) to the nearest tenth of a foot and make an accurate drawing of the track. c) Find the exact dimensions for x and y that will maximize the total area enclosed by the track. d) Repeat parts (a), (b), and (c) assuming that the one-mile length is measured on the outside edge of the track.
LINKING concepts... For Individual or Group Explorations Designing a Race Track An architect is designing a race track that will be 60 feel wide and have semicircular ends, as shown in the accompanying figure. The length of the track is to he one mile. measured on the inside edge of the track. a) Find the exact dimensions for x and y in the figure that will maximize the area of the rectangular center section. b) Round the answers to part (a) to the nearest tenth of a foot and make an accurate drawing of the track. c) Find the exact dimensions for x and y that will maximize the total area enclosed by the track. d) Repeat parts (a), (b), and (c) assuming that the one-mile length is measured on the outside edge of the track.
Solution Summary: The author calculates the dimensions of xandy in the figure that will maximize the area of the rectangular center section.
An architect is designing a race track that will be 60 feel wide and have semicircular ends, as shown in the accompanying figure. The length of the track is to he one mile. measured on the inside edge of the track.
a) Find the exact dimensions for x and y in the figure that will maximize the area of the rectangular center section.
b) Round the answers to part (a) to the nearest tenth of a foot and make an accurate drawing of the track.
c) Find the exact dimensions for x and y that will maximize the total area enclosed by the track.
d) Repeat parts (a), (b), and (c) assuming that the one-mile length is measured on the outside edge of the track.
ے ملزمة احمد
Q (a) Let f be a linear map from a space X into a space Y and (X1,X2,...,xn) basis for X, show that fis one-to-
one iff (f(x1),f(x2),...,f(x) } linearly independent.
(b) Let X= {ao+ax₁+a2x2+...+anxn, a;ER} be a vector space over R, write with prove a hyperspace and a
hyperplane of X.
مبر خد احمد
Q₂ (a) Let M be a subspace of a vector space X, and A= {fex/ f(x)=0, x E M ), show that whether A is
convex set or not, affine set or not.
Write with prove an
application of Hahn-Banach theorem.
Show that every singleton set in a normed space X is closed and any finite set in X is closed (14M)
Let M be a proper subspace of a finite dimension vector space X over a field F show that
whether: (1) If S is a base for M then S base for X or not, (2) If T base for X then base for M
or not.
(b) Let X-P₂(x) be a vector space over polynomials a field of real numbers R, write with L
prove convex subset of X and hyperspace of X.
Q₂/ (a) Let X-R³ be a vector space over a over a field of real numbers R and
A=((a,b,o), a,bE R), A is a subspace of X, let g be a function from A into R such that
gla,b,o)-a, gEA, find fe X such that g(t)=f(t), tEA.
(b) Let M be a non-empty subset of a space X, show that M is a hyperplane of X iff there
Xiff there
exists fE X/10) and tE F such that M=(xE X/ f(x)=t).
(c) Show that the relation equivalent is an equivalence relation on set of norms on a space
X.
Q/(a)Let X be a finite dimension vector space over a field F and S₁,S2CX such that S₁SS2. Show that
whether (1) if S, is a base for X then base for X or not (2) if S2 is a base for X then S, is a base for X or not
(b) Show that every subspace of vector space is convex and affine set but the conevrse need not to be true.
allet M be a non-empty subset of a vector space X over a field F and x,EX. Show that M is a
hyperspace iff xo+ M is a hyperplane and xo€ xo+M.
bState Hahn-Banach theorem and write with prove an application about it.
Show that every singleten subset and finite subset of a normed space is closed.
Oxfallet f he a function from a normad roace YI
Show tha ir continuour aty.GYiff
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