Engineering AnalysesENM 503
Methods and Models
Midterm Review
Spring 2012
1. Find the Fahrenheit temperature 10 degrees more than 1.5 times its equivalent in Celsius.
1.5C + 10 = 9C/5 + 32 => -0.3C = 22 => C = -73.333°
2. If an automobile averaged 40 miles per (mph) to A and returned at 60 mph, what was its average speed? Use an example of 120 miles
40 mph consumes 3 hours and 60 mph consumes 2 hours for a total of 5 hours and 120 + 120 = 240 miles or 240/5 = 48 mph.
3. Polynomial x4 - 9x3 + 21x2 + x -30 = 0 has 4 real roots. One is -1. What are the other 3?
4. A right triangle in the first quadrant is bounded by lines y = 0, y = x, and
y = -x + 5. Find its area.5
5
5. Find the solution set for |3x – 5| > 1.
-1 <= 3x – 5 <= 1 => 4/3 <= x <= 2 => Solution Set is (-, 4) U (2, )
6. The average of the test scores of a class of p students is 70 and the average of the test scores of a class of n students is 92. When the scores of both classes are combined, the average score is 86. What is the value of p/n?
86 = (70p + 92n)/ (p + n) or 86p + 86n = 70p + 92n
17p = 6n => p/n = 6/17.
7. Multiply the 2 matrices [1 2; 3 6] * [ 4 7; 2 0]
8. Find the inverse of matrix [1 5 ; 8 7 ] by attaching the unit matrix and performing elementary row operations.
9. Solve mentally for w and z: 1w – 2z = 3
2w + 2z = - 6
10.The solution to the general quadratic equation ax2 + bx + c = 0 is -3, 2.
Find a, b and c.
11. Find the maximum optimal solution given the constraints
3X1 + 4X2 24
5X1 + 3X2 29
Xi 0 29/3
for the following objective functions: 6
a. Z = 2X1 + 5X2;
b. Z = 23X1 + 5X2;
c. Z= 23X1 + 23X2.29/5 8
12. A man has a rope 180 feet long that he wishes to cut into three parts in the ratio of 2:3:4.
How long in feet will each piece of the rope be?
13. A straight line passes through the points (2,5) and (1,7). Write the equation of the line perpendicular to it.
14. 643/37= ?
15. Find annual payment for a 5-year loan of $20,000 at 5% compound interest.
16. The perimeter of a rectangle is 20 inches and one side is 4 inches. What is its area?
17. Given 3 points (2, 5), (4, 7), and (5, 5), find the area of the triangle formed.
Use Heron's formula (sqrt (* s (- s a)(- s b) (- s c)))Ans. 3
18. Solve for x: log10 x3 – 2 log10x = 2
19. If f(x) = 3x2 + 5x – 7, then f(-3) = ?
20. The following system of equations has how many solutions?
2x + 3y = 12
3x + 2y = 12
21. Log3 1237 = logx1237 implies that x = ______.
Log75043 = logx1237 implies that x = ______.
22. Write the equation for the y-axis, the x-axis, and the xy axes.
23. Graph {(x, y): |x| 12, |y| 10.
24. Find the inverse of f(x) = 5x – 3. Compute f(f(x)).
25. How many months to double your investment at a compound interest
rateof 5.2%?
26. What is the compound rate of interest for the cash flow below?
$2810.65Hint: Let x = 1 + i and resolve at year 2
2810.65x2 - 1000x – 2000 = 0
$1000 $2000
1 2
27. Write the dual of and solve by simplex: Minimize Z = 3x1 + 5x2
subject to -10x1 + 20x2 40
-20x1 + 20x2 80
xi 0
28. The rate of change of the area of a circle with respect to its diameter when the circumference of the circle is 5 is .
29. Find the inverse of matrix [5 -1; 1 0] by adjoining the identity and using elementary row operations.
30. If f(1) = 1 and f(x) = f(x - 1) + 2 for all x, then f(3) = .
31. Where does matrix [1 -1; 4 2] send vector [2, 5]?
32. Which is logically equivalent to "If not A then not B?"
a) If not B then not Ab) If not B, then Ac) If B then A
d) If A, then Be) If A, then not B.
33. Write NAND and NOR gates for A B inputs and C output.
A
B C
34. A triangle has sides 6x – 8y + 48 = 0, 5x + 4y - 20 = 0, and x – 10y + 1 = 0. Find equations for the altitudes.
35. xcos + y sin - k = 0 is an equation of a line.
Find the distance from the 3x + 4y – 15 = 0 and the point (4, 5).
|3(4)+ 4(5) – 15| / (32 + 42)1/2 = 17/5
36. Does the line 5x + 4y = 20 intersect the circle x2 + y2 = 9?
37. Write 497 in base 3 notation. Repeat for base 16. 200102
38. Base 7 notation 160061 is base 10 number______31256
39.
40. The set of vectors orthogonal to [1 -2 1] is
a) [x y 2y – x]b) [x y –x– y]c) [x y d) [x y ]
41. If f(x) = 2x3 – 2x and g(x) is its inverse, find g'(5).
y = f(x) => y – f(x) = 0; Implicit differentiate
dy - f'(x)dx = 0 => dx/dy = 1/f'(x) = 1/dy/dx
dy/dx = 6x2 – 2 = g'(x) and g'(5) = 148.
42. Find the maximum area enclosed in a rectangular fence with perimeter 100 units.
43. Write the truth table to show that the intersection of (NOT A) AND (NOT B) is equivalent to NOT(A OR B).
43. Is there a solution to the equation x = Ln x?
44. Write the power set (all subsets) of the set {1 2 3}. How many?
45. Show that y = x3 – 5x + 1 has slope zero for at least one x in (1, 2).
46a. Find the compound monthly payment on a 30-year loan of $500,000 if the APR is 12%.
b. What was the principal reduction for the 137th payment?
c. After 150 payments, the balance is $______.
d. After the final payment, the total interest paid was $ ______.
47. Write g(x) in terms of f(x) = 3 - 2x for
a) g(x) = 2f(2x) =
b) g(x) = f-1(x) =
c) If g(x) = 5x2, then (f g)(x) =
e) Given that f(x) and g(x) are inverse functions.
Then (f g)(x) – (g f)(x) = ______.
48. What are the 5 algebraic operators?
+ - * / and root extraction
49. Write the fundamental theorem of arithmetic. Unique prime factorization
50. Write the fundamental theorem of algebra. nth degree polynomials have n roots over the complex field.
51. Write the fundamental theorem of calculus. Differentiation & Integration
52. Write the primes using base 6 arithmetic and note the endings.
2 3 5 11 15 21 25 31 35
53. Write the Pythagorean Theorem.
54. Two trains on a collision course are traveling at 80 mph and 100 mp. How far apart are they at 1 minute before impact?
55. What is (¾ of ½ of 25) squared?
56. S(x, y) = (y-5, x + 5) T(x, y) = (x – y, y – x)
Find a) (S + T,b) 2S – T(x, y)
c) (ST)(x, y)
d) (TS)(x,y)
57. Write 237 in hexadecimal.
58. Evaluate the following determinant.
= 2
59. Find the time to make the 7th and the 8th item and the learning rate and slope/
Unit Hours Cumulative
1 1000.00 1000.00
2 760.00 1760.00
60. Find the eigenvalues and eigenvectors for matrix [1 2; 4 3].
61. A manufacturer can buy items at $1.25 or make them at $0.75 apiece. The equipment to make cost $2000. Find the breakeven point.3 647.28 2407.28
4 577.60 2984.88
5 528.76 3513.64
6 491.93 4005.58
7 462.81 4468.39
8 438.98 4907.36
9 418.97 5326.34
10 401.86 5728.19
11 386.97 6115.17
12 373.87 6489.04
.75Q + 2000 = 1.25Q => Q = 4000
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