MATHEMATICS COLLEGE

On 1st September 2014 there were 5400 trees planted in a wood.On 1st September 2015, only 5184 of these trees were still alive.
It is assumed that the number of trees still alive is given by N = art
where / is the number of trees still alive t years after 1st September 2014.
a) Write down the value

c) Show that on 1st September 2040
the number of trees still alive is predicted
o have decreased by over 65% compared
with September 2014.

b) Show that r = 0.96

Answers

Answer 1

Answer:

1. a = 5400

2. r = 0.96

3. Percentage decrement = 65.4%

Step-by-step explanation:

Given

N = ar^t

Solving (a): Write down the value of a

a implies the first term

And from the question, we understand that the initial number of trees is 5400.

Hence,

a = 5400

Solving (b): Show that r = 0.96

Using

N = ar^t

When a = 5400, t = 1 i.e. the first year and N = 5184

Substitute these values in the above expression

5184 = 5400 * r¹

5184 = 5400 * r

5184 = 5400r

Solve for r

r = 5184/5400

r = 0.96

Solving (c): Show that the trees has decreased by over 65% in 2040

First, we need to calculate number of years (t) in 2040

t = 2040 - 2014

t = 26

Substitute 26 for t, 5400 for a and 0.96 for r in N = ar^t to get the number of trees left

N = 5400 * 0.96^26

N = 1868.29658019

N = 1868 (approximated)

Next, we calculate the percentage change as thus:

%Change = (Final - Initial)/Initial * 100%

Where the initial number of trees =5400 and final = 1868

%Change = (1868 - 5400)/5400 * 100%

%Change = -3532/5400 * 100%

%Change = -3532%/54

%Change = -65.4%

The negative sign indicates a decrements or reduction.

Hence, percentage decrement = 65.4% and this is over 65%

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Look at the set of ordered pairs.{(4, 6), (−7,−15), (13, 15), (−21, 8), (?, ?)}

Which of the following could replace the missing ordered pair to make the set not a function?

(−2, 4)
(21, 9)
(−7, 15)
(6, 8)
(13,−15)

Answers

The ordered pair that could replace the missing ordered pair to make the set not a function is given as follows:

(−7, 15).

When does a set of ordered pairs represent a function?

A set of ordered pairs represents a function if every input (first component of the ordered pair) is associated with exactly one output (second component of the ordered pair). In other words, if no two ordered pairs in the set have the same first component but different second components.

Hence the ordered pair (-7,15) would make the relation not a function, as the set already has the ordered pair (-7,-15), in which the input 7 is already mapped to an output of -15, hence it cannot be mapped to an output of 15.

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The formula for the volume of a cylinder is V 7r²h, where r is the radius of the cylinder and his the
height of the cylinder. Rewrite the formula to solve for h in terms of r and V.
G

Answers

Final answer:

To solve for the height of a cylinder in terms of its radius and volume, rewrite the formula as h = V/(7r²).

Explanation:

To rewrite the formula to solve for h in terms of r and V, we need to isolate h on one side of the equation. The given formula is V = 7r²h. To isolate h, we divide both sides of the equation by 7r². This gives us the formula: h = V/(7r²). This formula allows us to calculate the height of a cylinder given its radius and volume.

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We play a card game where we receive 13 cards at the beginning out of the deck of 52. we play 50 games one evening. for each of the the following random variable identify the name and parameters of the distribution: a) The number of aces I get in the first game b) The number of games in which I recieve at least one ace during the evening c) The number of games in which all my cards are from the same suit d) The number of spades I receive in 5th game

Answers

The answer & explanation for this question is given in the attachment below.

Final answer:

The number of aces in the first game and the number of spades in the 5th game follow a Hypergeometric Distribution while the number of games receiving at least one ace can be modeled by a Binomial distribution. The event of all cards being from the same suit can be thought of as a Uniform distribution.

Explanation:

a) The number of aces you get in the first game follows a Hypergeometric Distribution. In such a distribution, you are drawing cards without replacement. The parameters are N=52 (the population size), K=4 (the number of success states in the population i.e., the number of aces in a deck), and n=13 (the number of draws).

b) The number of games in which you receive at least one ace can be modeled by a Binomial distribution. Each game you play (out of 50) is a single trial, with the probability of success (getting at least one ace) being the same for every trial. The parameters are n=50 (the number of trials/games) and p (the probability of getting at least one ace).

c) The likelihood of all your cards being from the same suit in a game is heavily reliant on chance, can be modeled as a Uniform distribution given its rare occurrence. Essentially, the parameters would be minimum = 0 and maximum = 1. However, determining the parameters would require calculation of the specific probabilities, which is complex due to the nature of the game.

d) The number of spades you receive in the 5th game also follows a Hypergeometric distribution, similar to the situation in the first game. The parameters in this case are N=52, K=13 (number of spades in a deck), and n=13 (the number of drawn cards).

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