University:
Carleton College
Course:
MATH 111 | lntroduction to Calculus
Academic year:

2024
Views:

77

Pages:

2
Author:

Alecai8y

Question 1 Solve for y: (7y + 3)/2 = (8y + 1)/6 + 23 Simplify your answer as much as possible. Answer We are given: (7y + 3)/2 = (8y + 1)/6 + 23 Multiply both sides by the LCD which is 6: ((7y + 3)/2)(6) = ((8y + 1)/6 + 23)(6) 21y + 9 = (8y + 1) + 138 21y + 9 = 8y + 139 Subtract 9 from both sides: 21y = 8y + 130 Subtract 8y from both sides: 13y = 130 Divide both sides by 13: y = 10 Question 2 Insert one of the symbols << or >> to make the following statements true. a) If x > 5, then x + 8 __ 3. b) If x < -3, then -5x + 18 __ 3. Answer We rewrite the inequality so that it matches the left side of the "then" part. a) By adding 8 to each side, we have: x + 8 > 5 + 8 x + 8 > 13 So, insert >. b) Multiply both sides by -5. Recall that when we multyply of divide both sides of an inequality by a negative number, we change the direction of the inequality: -5x > 15 Add 18 to both sides: -5x + 18 > 3 So, insert <. Question 3 Take two cards from a deck without replacement. What is the probability of getting a red queen and then another red card? Answer There are 2 red queens (hearts and diamonds) in a standard deck of cards, the probability of drawing a red queen is: 2/52 = 1/26 There isn't a replacement, so that leaves a total of 13 + 12 = 25 red cards out of the remaining 51 cards so the probability that the next crad drawn is a red card is: 25/51 Therefore, the probability of getting a red queen and then another red card is the product of the probabilities: (1/26) * (25/51) = 25/1326 Question 4 Ten red cards and ten black cards are placed in a bag. You choose one card and then another without replacing the first card. What is the probability that the first card will be red and the second card will be black? Answer The events are dependent because the outcome of drawing the first card affects the outcome of the next draw. With two dependent events A and B, we also apply the Multiplication Rule: P(A then B) = P(A) * P(B after A) The probability that the first draw is a red card is 10/20 = 1/2. The probability that the second draw is a black card is now 10/19 (since there is no replacement). Hence, the probability of first drawing red card then a black is: P(red then black) = P(red) * P(black after red) P(red then black) = (1/2) * (10/19) P(red then black) = 5/19 Question 5 Suppose that f is an exponential function with percentage growth rate of 5% and that f(0) = 4. Find a formula for f(x). Answer Use the exponential equation: f(x) = a(1+r)^x where a is the initial value and r is the growth rate/decay. Given f(0) = 4, then a = 4. Since we are given a growth rate, then r = 0.05: f(x) = 4(1+0.05)^x f(x) = 4(1.05)^x