SOLUTIONS TO HOMEWORK #1 (THE NON-GRAPHICAL SOLUTIONS ANYHOW) --------------------------------Warren D. Smith----Math 75---- PROBLEM 1. It's easiest to derive F(C) first. Since linear function F = a*C + b. But what are a and b? we know 32 = a*0 + b 212 = a*100 + b so from the 1st equation b=32, a = (212-32)/100 = 9/5. so --------------------------------------- ANSWER(b) F = (9/5) C + 32. --------------------------------------- ANSWER(e) The slope is (9/5). --------------------------------------- Now to find C(F), solve for C by subtracting 32 from both sides of answer b: F-32 = (9/5) C and multiply by 5/9 to get: --------------------------------------- ANSWER(a) C = (5/9) (F-32). --------------------------------------- To find when C=F, we must solve F = (9/5) F + 32 for F. Subtract (32+F) from both sides: -32 = (9/5) F - F = (9/5 - 5/5) F = (4/5) F and multiply -32 = (4/5) F by 5/4 to get -32(5/4) = F which is: --------------------------------------- ANSWER(c) -40 = F = C. --------------------------------------- --------------------------------------- ANSWER(d) [plot of line thru (0,32) and (100,212).] --------------------------------------- PROBLEM 2. If the chessboard had N squares the answer (call it A(N)) would be N - 1 ----- \ k A(N) = ) 2 / ----- k = 0 compute A(0) = 0 A(1) = 1 A(2) = 1+2 = 3 A(3) = 1+2+4 = 7 A(4) = 1+2+4+8 = 15 A(5) = 1+2+4+8+16 = 31 [Common errors: a lot of people could not even get this far without screwing up, or had some insane formula that did not agree with these firast few value. Sanity checking would have spotted!] see a pattern? Seems to obey the formula N A(N) = 2 - 1. Prove by induction this keeps working forever: N N N N N+1 A(N+1) = A(N) + 2 = 2 - 1 + 2 = 2(2 ) - 1 = 2 -1. ^ ^ | | If A(N) were correct, This proves the formula works for what would A(N+1) be? A(N+1) if it works for A(N) It would be A(N) plus one N extra term in the sum, namely 2. Q.E.D. So --------------------------------------- 64 ANSWER(a): A(64) = 2 - 1 grains of rice total. Exactly. --------------------------------------- 19 ANSWER(b): = 1.84 X 10 to 1% accuracy. --------------------------------------- Playing with rice I see about 30 grains will fit in 1 cc (1 cubic centimeter) so the total volume of all this rice is about this/30 ccs, i.e. about 17 6 X 10 cc. That is a lot. It is the volume of a cube about 8 km on a side, 17 3 since [8km*(100 cm/meter)*(1000 meter/km)]^3 = 5 X 10 cm. This is about the size of Mount Everest. It is about 500 times more weight than the weight of all the humans in the world put together. It would bankrupt the king of the world. But it is still MUCH LESS than the size of the whole planet. PROBLEM 3. 5 * 8 * sin(20 degrees)/2 = 5*8*sin(20*3.1416/180)/2 = 6.84 since Area = Base * Height/2, where Base=8, Height=5 sin(20 degrees) [from defn of sin] [Common errors: forgetting the /2 (triangle is HALF the area of rectangle of same base and height) computing sin(20 radians) not sin(20 degrees) by misusing calculator remember: pi radians = 180 degrees thinking this is a right-triangle forgetting the definition of sin(angle)] Sanity check (draw the triangle fairly accurately, look at it or measure it to estimate the height and the area) would have prevented ALL these errors! PROBLEM 4. [Plots.] The domains are, the whole real line, although in the case of (c) x = sqrt(2) and x = -sqrt(2) need to be omitted to avoid division by 0. The range of (a) is: y>=2. Of (b) is: y <= 1/2. Of (c) is: (0, infinity) U (-infinity, -1/2]. The plots look like: (a) parabola, min at (0,2) goes up on both sides. Only this is concave-up. (b) bell shaped curve, max at (0, 1/2), asymptotes to y=0 as x --> infinity or as x --> -infinity. Near the max this curve is concave-down, but when |x| is large it is concave-up. (c) made of 3 curves: I: curve that asymptotes to y=0 as x --> -infinity and to y=+infinity as x goes to -sqrt(2) from the left. Concave-up. II: curve that asymptotes to y=-infinity as x goes to -sqrt(2) from the right; has a max at (0, -1/2); and asymptotes to y=-infinity as x goes to +sqrt(2) from the left. Concave-down. III: curve that asymptotes to y=0 as x --> +infinity and to y=+infinity as x goes to +sqrt(2) from the right. Concave-up. PROBLEM 5. ANSWER(a) N 1000 * (0.7) is wealth after N losing years. 5 N=5 gives: 1000 * (0.7) = 168.07. N Now after N winning years wealth = 168.07 * (1.3) N=5 gives: 624.03. Why is this not 1000 back again? Because 30% of a big number (before the loss) is bigger than 30% of a small number (before the gain). So the gains do not make up for the losses. PROBLEM 6. 2 S + S = 7 solutions to this quadratic 2 S + S - 7 = 0 are -1 +- sqrt(29) S = -------------- 2 and we had better pick the +, not the -, sign on the sqrt to avoid a negative S as an answer (lengths are not negative), so the only valid answer is sqrt(29) - 1 S = ------------- = 2.19 to 1% accuracy. 2 [common errors: some people thought the area of an S by 1 rectangle was S+1 or 2S+2. No, it is height TIMES width = 1 TIMES S !! Sanity check: try S=1, would have spotted these errors!] Other people did wrong arithmetic. Sanity check: take whatever final answer S you get, compute (S+1)S, and if don't get 7, know insane! Other people forgot the quadratic formula.]