Math Answers

A boy is flying a kite at a height of 150 ft. If the kite moves horizontally away from the boy at the rate of 20 ft/sec, how fast is the string being paid out when the kite is 250 ft from him?

A spherical snowball with an outer layer of ice melts, so that the radius of the snowball decreases at the rate of 1/5 cm/sec. Find the rate at which the volume decreases when the diameter is 50 cm.

Registrants at large convention are offered 9 sight seeing tours on each of 3 days. In how many ways can a person arrange to go on one of these tours?

Suppose we roll two dice and let X be the maximum of the two numbers obtained and let Y be their minimum. Write down the distribution tables of X, Y and Z = (X, Y ).

Let A, B, C, D denote, respectively, art, biology, chemistry, and drama courses.

Find the number N of students in a dormitory given the data:

12 take A, 5 takeAand B, 4 takeB and D, 2 take B, C,D,

20 take B, 7 takeAand C, 3 takeC and D, 3 take A, C,D,

20 take C, 4 takeAand D, 3 take A, B,C, 2 take all four,

8 take D, 16 takeB and C, 2 take A, B, D, 71 take none.

1.     The following formulas have been abbreviated based on the common abbreviation rules. Follow the steps below and translate the formulas into good English.

·       Step 1: Re-add the omitted brackets.

·       Step 2: If necessary, convert them into some other logically equivalent formula

so as to make it more readable. Write out the rule(s) you use for conversion.

·       Step 3: Translate the formulas into `good' English. Try to make your translation as brief/understandable as possible. (For instance, `John and Bill are coming' is better than `John is coming and Bill is coming.')

p: John wants to come to the class.

q: John will come to the class today.

r: John audits the class.

s: John is enrolled in the class.

Hint:

`No matter whether John is going or not, I'm going.' is the translation for (j à i) ^ (⌐j à i),

in which j = John is going, i = I'm going.)

[R] Check the convergence of the sequence defined by 𝑢𝑛+1 = 𝑎/ 1+𝑢𝑛 where 𝑎 > 0, 𝑢1 > 0.

Use the method of variation of parameters to solve the differential equation:

y"-2y'+y=xe^xtan-1x

State TRUE or FALSE justifying your answer with proper reason.

a. 2𝑛^2 + 1 = 𝑂(𝑛^2 )

b. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 )

c. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 log 𝑛)

d. 3𝑛^2 + √𝑛 = 𝑂(𝑛 + 𝑛√𝑛 + √𝑛)

e. √𝑛 log 𝑛 = 𝑂(𝑛)

solve the following recurrence relations

a. 𝑇(𝑛) = 𝑇( 𝑛/4) + 𝑇( 𝑛/2 ) + 𝑛^2

b. T(n) = T(n/5) + T(4n/5) + n

c. 𝑇(𝑛) = 3𝑇( n/4 ) + 𝑐𝑛^2

f. 𝑇(𝑛) = (𝑛/𝑛−5) * 𝑇(𝑛 − 1) + 1

g. 𝑇(𝑛) = 𝑇(log 𝑛) + log 𝑛

h. 𝑇(𝑛) = 𝑇 (𝑛^ 1/ 4) + 1

i. 𝑇(𝑛) = 𝑛 + 7 √𝑛 ∙ 𝑇(√𝑛)

j. 𝑇(𝑛) = 𝑇 ( 3𝑛/4 ) + 1/root(n)