Answer:
600ft x 1200ft
Step-by-step explanation:
Use derivative optimization to find the maximum area.
I'll call the two same sides "a", and the one different side "b"
The maximum perimeter (including 3 sides) is 2400 ft. so,
2400 = 2a + b
The area is length × width. so,
A = ab
Using substitution to combine the equations,
A = a × (2400 - 2a)
A = -2a² + 2400a
Find the maximum of A by finding the zeros of its derivative.
dA = -4a +2400
0 = -4a + 2400
The maximum occurs at a = 600
Substitute in the perimeter equation to find b.
2400 = 2(600) + b
b = 1200
600 x 1200
Is anyone wanna help me?
If you help me you will earn 8+ pts.
( i need all questions answer )
Answer:
See the answers bellow
Step-by-step explanation:
For 51:
Using the definition of funcion, given f(x) we know that different x MUST give us different images. If we have two different values of x that arrive to the same f(x) this is not a function. So, the pair (-4, 1) will lead to something that is not a funcion as this would imply that the image of -4 is 1, it is, f(-4)=1 but as we see in the table f(-4)=2. So, as the same x, -4, gives us tw different images, this is not a function.
For 52:
Here we select the three equations that include a y value that are 1, 3 and 4. The other values do not have a y value, so if we operate we will have the value of x equal to a number but not in relation to y.
For 53:
As he will spend $10 dollars on shipping, so he has $110 for buying bulbs. As every bulb costs $20 and he cannot buy parts of a bulb (this is saying you that the domain is in integers) he will, at maximum, buy 5 bulbs at a cost of $100, with $10 resting. He can not buy 6 bulbs and with this $10 is impossible to buy 0.5 bulbs. So, the domain is in integers from 1 <= n <= 5. Option 4.
For 54:
As the u values are integers from 8 to 12, having only 5 possible values, the domain of the function will also have only five integers values, With this we can eliminate options 1 and 2 as they are in real numbers. Option C is the set of values for u but not the domain of c(u). Finally, we have that 4 is correct, those are the values you have if you replace the integer values from 8 to 12 in c(u). Option 4.
The population P of a certain city can be modeled by P=19000e0.0215t where T represent the number of years since 2000. When t=1 the year is 2001 when t =2 the year is 2002 based in yhis model in what year will the population reach 40,000
Answer:
Step-by-step explanation:
The population P of a certain city can be modeled by
P=19000e0.0215t
where T represent the number of years since 2000.
When t=1 the year is 2001
when t =2 the year is 2002. This means that
When t=0 the year is 2000
To determine the year when the population will be 40,000 , we will substitute P = 40,000 and solve for t. It becomes
40000 = 19000e^0.0215t
40000 / 19000 = e^0.0215t
2.105 = e^0.0215t
Take ln of both sides
0.7443 = 0.0215t
t = 0.7443/0.0215
t = 34.62
Approximately 35 years
So the year will be 2035
In 1990, 75% of all high school seniors had jobs. In 2010, 40% of all seniors had jobs. If there were 50 more seniors in 2010 than 1990 and there were a total of 403 jobs between both years, how many seniors were there in 1990? How many seniors had jobs in 2010
Answer:
There were 333 seniors in 1990
153 seniors had jobs in 2010.
Step-by-step explanation:
Let the number of high school seniors in 1990 = x
75% of x had jobs in 1990
In 2010 there were 50 more seniors. That is, the number of seniors is x+50.
40% of (x+50) had jobs in 2010.
Total number of jobs in both years = 403.
\[0.75 * x + 0.4 * (x + 50) = 403\]
\[0.75 * x + 0.4 * x + 20 = 403\]
=> \[1.15 * x = 403 - 20\]
=> \[1.15 * x = 383\]
=> \[x = 383/1.15\]
=> \[x = 333\]
Number of seniors having job in 2010 = 0.4 * (333 + 50) = 0.4 * 383 = 153.2 = 153(approx)
Which of the following is a characteristic of an experiment where the binomial probability distribution is applicable?a. The experiment has at least two possible outcomes.b. Exactly two outcomes are possible on each trial.c. The trials are dependent on each other.d. The probabilities of the outcomes changes from one trial.
Answer:
b. Exactly two outcomes are possible on each trial.
Step-by-step explanation:
The correct answer is option B: Exactly two outcomes are possible on each trial.
We can define the binomial probability in any binomial experiment, as the probability of getting exactly x successes for n repeated trials, that can have 2 possible outcomes.
The binomial probability distribution is applicable when an experiment has exactly two outcomes. These trials are independent and the probabilities of the outcomes remain the same for each trial, referred to as Bernoulli trials. Using this distribution, we can calculate mean and standard deviation for the function.
Explanation:The characteristic of an experiment where the binomial probability distribution is applicable is when exactly two outcomes are possible on each trial, referred to as 'success' and 'failure.' These are termed as Bernoulli trials for which binomial distribution is observed. In this situation, 'p' denotes the probability of a success on one trial, while 'q' represents the failure likelihood. The trials are independent, which means the outcome of one trial does not affect the results of subsequent trials. Moreover, the probabilities of the outcomes remain constant for every trial. The random variable X signifies the number of successes in these 'n' independent trials. The mean and standard deviation can be calculated using the formulas µ = np and √npq, respectively.
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Addison uses up all the dog food in 6 days. Each day, she feeds her dogs
the same amount of food. Between which two numbers is the number of
cups of food she feeds her dogs each day? There are 31 cups of dog food.
A. O and 1
B. 1 and 2
C. 5 and 6
D. 6 and 7
Answer:
Step-by-step explanation:
She uses all the food to feed the dogs in 6 days. She feeds the with an equal amount of food in 6 days. This means that whatever amount of food she buys, the least amount she would feed the dogs with is 1/6 of that amount.
There are 31 cups of dog food. This means that the amount she would feed them with each day is
1/6 × 31 = 5.1 cups
So the least amount is 5 cups
Therefore,
The number of cups she feeds her dogs with would be between 5 and 6 cups.
The probability that a randomly selected person has high blood pressure (the event H) is P(H) = 0.4 and the probability that a randomly selected person is a runner (the event R) is P(R) = 0.4. The probability that a randomly selected person has high blood pressure and is a runner is 0.1. Find the probability that a randomly selected person either has high blood pressure or is a runner or both.
Answer:
0.7 is the probability that a randomly selected person either has high blood pressure or is a runner or both.
Step-by-step explanation:
We are given the following information in the question:
Probability that a randomly selected person has high blood pressure = 0.4
[tex]P(H) = 0.4[/tex]
Probability that a randomly selected person is a runner = 0.4
[tex]P(R) = 0.4[/tex]
Probability that a randomly selected person has high blood pressure and is a runner = 0.1
[tex]P(H \cap R) = 0.1[/tex]
If the events of selecting a person with high blood pressure and person who is a runner are independent then we can write:
[tex]P(H \cup R) = P(H) + P(R)-P(H\cap R)[/tex]
Probability that a randomly selected person either has high blood pressure or is a runner or both =
[tex]P(H \cup R) = P(H) + P(R)-P(H\cap R)\\P(H \cup R) = 0.4 + 0.4 -0.1 = 0.7[/tex]
0.7 is the probability that a randomly selected person either has high blood pressure or is a runner or both.
Final answer:
The probability that a randomly selected person either has high blood pressure or is a runner or both is 0.7.
Explanation:
To find the probability that a randomly selected person either has high blood pressure or is a runner or both, we use the formula for the probability of either event A or event B occurring, which is:
P(A or B) = P(A) + P(B) - P(A and B).
Given:
P(H) = probability of high blood pressure = 0.4
P(R) = probability of being a runner = 0.4
P(H and R) = probability of both high blood pressure and being a runner = 0.1
Using the formula, we plug in the given probabilities:
P(H or R) = 0.4 + 0.4 - 0.1 = 0.7.
So, the probability that a randomly selected person either has high blood pressure, is a runner, or both events occur is 0.7.
Suppose A is n x n matrix and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In. By Theorem 7, this shows that A must be Invertible.)
Theorem 7: An n x n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transfrms In into A-1.
Answer:
Remember, a homogeneous system always is consistent. Then we can reason with the rank of the matrix.
If the system Ax=0 has only the trivial solution that's mean that the echelon form of A hasn't free variables, therefore each column of the matrix has a pivot.
Since each column has a pivot then we can form the reduced echelon form of the A, and leave each pivot as 1 and the others components of the column will be zero. This means that the reduced echelon form of A is the identity matrix and so on A is row equivalent to identity matrix.
On a trip over Thanksgiving Break Jenn's family has to drive a total of 325 miles.. If there are 126 miles to go and the car has been driving for 3 hours, What is the speed? Explain how this relates to slope?
The speed is 66.33 miles per hour
Step-by-step explanation:
Given
Total distance = 325 miles
They still have 126 miles to go. In order to find the distance they have covered, we will subtract the remaining miles from the total
So,
[tex]Distance\ covered=325-126=199\ miles[/tex]
the car has been driving for 3 hours
So,
t= 3 hours
d = 199 miles
[tex]Speed=\frac{d}{t}\\=\frac{199}{3}\\=66.33\ miles\ per\ hour[/tex]
The speed is 66.33 miles per hour
Keywords: Speed, Distance
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What do i do to find EF in the trapezoid? I can't seem to figure it out.
Answer:
x=4
Step-by-step explanation:
EF is half of the top and bottom lines comined so you would make an equation:
5x+8/2=2x+6
5x+8=4x+12
x=4
ASAP PLZ!!! Select the correct answer. Solve for x. 2x2 − 4x = 0 A. 0, -4 B. 0, -2 C. 0, 2 D. 2, 4
Answer:
C
Step-by-step explanation:
2x²-4x=0
2x(x-2)=0
either x=0
or x-2=0
x=2
so x=0,2
How much difference do a couple of weeks make for birth weight? Late-preterm babies are born with 34 to 36 completed weeks of gestation. The distribution of birth weights (in grams) for late-preterm babies is approximately N(2750, 560).
1. What is the probability that a randomly chosen late-preterm baby would have a low birth weight (less than 2500 grams)? Round your answer to 4 decimal places.
2. What is the probability that a randomly chosen late-preterm baby would have a very low birth weight (less than 1500 grams)? Round your answer to 4 decimal places.
Answer:
a) 0.3277
b) 0.0128
Step-by-step explanation:
We are given the following information in the question:
N(2750, 560).
Mean, μ = 2750
Standard Deviation, σ = 560
We are given that the distribution of distribution of birth weights is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) P (less than 2500 grams)
P(x < 2500)
[tex]P( x < 2500) = P( z < \displaystyle\frac{2500 - 2750}{560}) = P(z < -0.4464)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 2500) = P(z < -0.4464) = 0.3277 = 32.77\%[/tex]
b) P ((less than 1500 grams)
P(x < 1500)
[tex]P( x < 1500) = P( z < \displaystyle\frac{1500 - 2750}{560}) = P(z < -2.2321)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 1500) = P(z < -2.2321) = 0.0128 = 1.28\%[/tex]
Final answer:
The probability of a late-preterm baby having a low birth weight (< 2500 grams) is approximately 32.74%, and the probability of a very low birth weight (< 1500 grams) is about 1.29%.
Explanation:
To answer the given questions, we will use the properties of the normal distribution, where the mean birth weight (μ) for late-preterm babies is 2750 grams, and the standard deviation (σ) is 560 grams. The distribution of birth weights is assumed to be normal (N(2750, 560)).
1. Probability of a Low Birth Weight (< 2500 grams)
To find the probability of a baby having a low birth weight (less than 2500 grams), we use the Z-score formula: Z = (X - μ) / σ, where X is the value of interest (2500 grams). Plugging in the values gives us Z = (2500 - 2750) / 560 = -0.4464. Using a Z-table or a normal distribution calculator, we find the probability corresponding to Z = -0.4464, which is approximately 0.3274. Therefore, the probability of a late-preterm baby having a low birth weight is about 0.3274 or 32.74%.
2. Probability of a Very Low Birth Weight (< 1500 grams)
To calculate the probability of a very low birth weight (less than 1500 grams), again we calculate the Z-score: Z = (1500 - 2750) / 560 = -2.2321. The probability of Z = -2.2321, referring to the Z-table or calculator, is extremely low, approximately 0.0129 or 1.29%. This indicates that the chances of a late-preterm baby having a very low birth weight is about 1.29%.
A stores having a sale on jelly beans and trail mix for 8 pounds of jelly beans and 4 pounds of trail mix the total cost is $25. For 3 pounds of jelly beans and 2 pounds of trailmix the total cost is $10. Find the cost for each pound of jelly beans and each pound of trailmix
Answer:each pound of jelly beans cost $2.5
Each pound of trailmix cost $1.25
Step-by-step explanation:
Let x represent the cost of one pound of jelly bean.
Let y represent the cost of one pound of trail mix.
A stores having a sale on jelly beans and trail mix for 8 pounds of jelly beans and 4 pounds of trail mix the total cost is $25. This means that
8x + 4y = 25 - - - - - - -1
For 3 pounds of jelly beans and 2 pounds of trailmix the total cost is $10. This means that
3x + 2y = 10 - - - - - - - - 2
Multiplying equation 1 by 3 and equation 2 by 8, it becomes
24x + 12y = 75
24x + 16y = 80
Subtracting,
- 4y = -5
y = - 5/ -4 = 1.25
Substituting y = 1.25 into equation 2, it becomes
3x + 2×1.25 = 10
3x + 2.5 = 10
3x = 10 - 2.5 = 7.5
x = 7.5/3 = 2.5
The stem-and-leaf plot shows kilometers walked by participants in a charity benifit walk. Use it to answer the questions. a. How many people participated in the walk? b. How many of the walkers traveled more than 14 Kilometers?
The plot is missing. I have written the stem-leaf plot below.
Answer:
(a) 35 participants
(b) 22 participants traveled more than 14 km.
Step-by-step explanation:
Given:
(a)
The stem-leaf plot is given as:
12| 3 3 6 7 9 9
13| 1 1 4 5 5
14| 0 0 2 3 3 8 8 9
15| 2 2 2 2 2 3 5 5 7
16| 4 5 5 9 9
17| 3 5
The total number of numbers on the leaf side gives the total number of participants.
So, the number of participants is equal to the total number of elements on the leaf part. The total number of elements on the leaf part is 35.
Therefore, the total number of participants is 35.
(b)
Given:
12| 3 3 6 7 9 9
13| 1 1 4 5 5
14| 0 0 2 3 3 8 8 9
15| 2 2 2 2 2 3 5 5 7
16| 4 5 5 9 9
17| 3 5
One element of the stem part and one part of leaf part is represented as:
12 | 3 means 12.3 km
So, the number of walkers traveling more than 14.0 km is the list of all numbers greater than 14.0 km. Let us write all the numbers which are greater than 14.0 km. The numbers are:
14 | 2 3 3 8 8 9 = 6 participants
15 | 2 2 2 2 2 3 5 5 7 = 9 participants
16 | 4 5 5 9 9 = 5 participants
17 | 3 5 = 2 participants
Therefore, the total number is the sum of all the above which is equal to:
= 6 + 9 + 5 + 2 = 22 participants
Consider the following functions:
f = {( -4, 2), (1, -1)} and g= {(0, 3), (1,2), (-3,-2), (-4,2)}
(f+g)(1)= ?
Answer:
1
Step-by-step explanation:
(f+g)(1) = f(1) +g(1) = -1 + 2 = 1
__
The pair (1, -1) in the definition of f tells you f(1) = -1.
The pair (1, 2) in the definition of g tells you g(1) = 2.
How am I supposed to solve this? I know how to do the equations but I don't know why there are only 2 places to put an answer for X and Y when there are 2 equations. Both equations have an X and a Y so that means there are 4 X and Y answers.
Answer:
x = 1
y = -4
Step-by-step explanation:
We will multiply the 2nd equation by 4 first:
4 * [-x + y = -5]
-4x + 4y = -20
Now we will add this equation with 1st equation given. Shown below:
4x + 3y = -8
-4x + 4y = -20
---------------------
7y = -28
Now, we can solve for y easily:
7y = -28
y = -28/7
y = -4
Now, we take this value of y and put it in 1st original equation and solve for x:
4x + 3y = -8
4x + 3(-4) = -8
4x - 12 = -8
4x = -8 + 12
4x = 4
x = 4/4
x = 1
So, this is the only solution to this problem ( 1 intersection point at x = 1 and y = -4)
Jason paid $15.50 for 3 slices of pizza and 2 burgers. Susan paid $20 for 1 slice of pizza and 4 burgers. Write a system of equation and then determine the cost of each slice of pizza and the cost of each burger.
Each slice of pizza costs $
Each burger costs $
Answer:
The answer to your question is: Burger = $4.45, Pizza = $2.2
Step-by-step explanation:
Jason = $15.5 for 3 slices of pizza + 2 burgers
Susan = $20 for 1 slice of pizza + 4 burgers
Pizza = p
burger = b
System of equations
Jason 3p + 2b = 15.5 (I)
Susan p + 4b = 20 (II)
Solve system by elimination
Multiply (II) by -3
3p + 2b = 15.5
-3p - 12b = -60
-10b = -44.5
b = -44.5/-10
b = $4.45
p + 4(4.45) = 20
p + 17.8 = 20
p = 20 - 17.8
p = 2.2
One slice of pizza costs $2.2 and one burger cost $4.45
Find f ′(x) for f(x) = ln(x^3 + e^4x).
SHOW WORK
Answer:
f(x) = ln(x^3 + e^4x) answer is explain in attachment .
Step-by-step explanation:
f(x) = ln(x^3 + e^4x) =(2x+3e³ˣ) 1/x² + e³ˣ
NO ONE KNOWS THIS ANSWER?
how does the graph f(x) =(x-8)^3+4 compare to the paren function g(x)=x^3?
Not multiple choice.
Answer:
Their intercepts are unique.
Explanation:
[tex]\displaystyle x^3 - 24x^2 + 192x - 508 = (x - 8)^3 + 4[/tex]
This graph's x-intercept is located at approximately [6,41259894, 0], and the y-intercept located at [0, −508].
[tex]\displaystyle g(x) = x^3[/tex]
The parent graph here, has both an x-intercept and y-intercept located at the origin.
I am joyous to assist you anytime.
Phil is going to store to buy a hat and a coat. The coat cost 3 times as much as the hat. His mom tells him that he cannot spend more than $120. What is the most he can spend for the coat?
Answer:
Step-by-step explanation:
Phil is going to store to buy a hat and a coat.
Let $x represent the cost of the cos of the hat. The coat cost 3 times as much as the hat. It means that the cost of the coat will be 3×x = $3x
His mom tells him that he cannot spend more than $120. Assuming he spends exactly $120. It means that
3x + x = 120
4x = 120
x = 120/ 4 = 30
The hat costs $30
The coat costs 30×3 = $90
The most he can spend on a coat is $90 since he cannot spend more than $120
From base camp, a hiker walks 3.5 miles west and 1.5 miles north. Another hiker walks 2 miles east and 0.5 miles south. To the nearest tenth of a mile how far apart are the hikers
Answer:
The hikers are 5.9 miles apart.
Step-by-step explanation:
Let O represents the base camp,
Suppose after walking 3.5 miles west, first hiker's position is A, then after going 1.5 miles north from A his final position is B,
Similarly, after walking 2 miles east, second hiker's position is C then going towards 0.5 miles south his final position is D.
By making the diagram of this situation,
Let D' is the point in the line AB,
Such that, AD' = CD
In triangle BD'D,
BD' = AB + AD' = 1.5 + 0.5 = 2 miles,
DD' = AC = AO + OC = 3.5 + 2 = 5.5 miles,
By Pythagoras theorem,
[tex]BD^2 = BD'^2 + DD'^2[/tex]
[tex]BD = \sqrt{2^2 + 5.5^2}=\sqrt{4+30.25}=\sqrt{34.25}\approx 5.9[/tex]
Hence, the hikers are 5.9 miles apart.
How many different license plates are possible if each contains 3 letters (out of the alphabet's 26 letters) followed by 2 digits (from 0 to 9)? How many of these license plates contain no repeated letters and no repeated digits?
Answer:
The number of ways the license plates contain no repeated letters and no repeated digits = 1,404,000
Step-by-step explanation:
The total number of alphabet's = 26
The total number of digits = 10 (0 - 9)
The number plate contains 3 letters followed by 2 digits.
The number of ways the license plates contain no repeated letters and no repeated digits.
The number of ways first letter can be filled in 26 ways.
The number of ways second letter can be filled in 25 ways.
The number of ways third letter can be filled in 24 ways.
The number of ways the first digit can be in 10 ways
The number of ways the second digit can be in 9 ways.
The number of ways the license plates contain no repeated letters and no repeated digits = 26 × 25 × 24 × 1 0 × 9
The number of ways the license plates contain no repeated letters and no repeated digits = 1,404,000
Omar and Mackenzie want to build a pulley system by attaching one end of a rope to their 8-foot-tall tree house and anchoring the other end to the ground 28 feet away from the base of the treehouse. How long, to the nearest foot, does the piece of rope need to be?
a. 26ft
b. 27ft
c. 28ft
d. 29ft
Answer:
d. 29ft
Step-by-step explanation:
Using pythagoras theorem which states that
a^2 + b^2 = c^2
where a and b are the opposite and adjacent sides of a right angled triangle and c is the hypotenuse side
From the attached image, Let the length of the rope be x
[tex]x^{2} = 8^{2}+ 28^{2} \\[/tex]
[tex]x^{2} = 64 + 784[/tex]
[tex]x^{2} = 848[/tex]
[tex]x =\sqrt{848}[/tex]
[tex]x = 29.12[/tex]
x≈29
f(x) = x2 + 1 g(x) = 5 – x
(f – g)(x) =
A. x2 + x – 4
B. x2 + x + 4
C. x2 – x + 6
D. x2 + x + 6
Answer:
c
Step-by-step explanation:
Which binomial expressions are factors of 2x3+5x2−x−6?
Answer:
x − 1
x + 2
Step-by-step explanation:
f(x) = 2x³ + 5x² − x − 6
Plug in the zero of each binomial. If that zero is also a zero of f(x), then the binomial is a factor of f(x).
f(1) = 0
f(-1) = -2
f(2) = 28
f(-2) = 0
The correct options are option A and option D that is the factors of given binomial expressions are x - 1 and x + 2.
What is binomial factor ?
Binomial factors are polynomial factors that have exactly two terms. Factoring a polynomial is the first step to finding its roots.
The given binomial expression is equal to :
2x³ + 5x² - x - 6
We know that factor theorem states that if [tex]x_{1}[/tex] , [tex]x_{2}[/tex], --- --- --- --- , [tex]x_{n}[/tex] are roots of a function , them (x - [tex]x_{1}[/tex]) , (x - [tex]x_{2}[/tex]) , --- --- --- --- , ( x - [tex]x_{n}[/tex]) are the factors of given binomial expression.
In the given question :
f(x) = 2x³ + 5x² - x - 6
If we put x = 1 , then we get f(x) = 0 . This implies the binomial factor will be (x - 1).
So ,
(ax² + bx + c ) ( x - 1) = 2x³ + 5x² - x - 6
ax³ + (b-a) x² + (c - b) x - c = 2x³ + 5x² - x - 6
If we compare both sides we get :
a = 2 , b - a = 5 , c - b = 1 and c = 6
So , b = 7
We know that general form is given by :
ax² + bx + c
or
2x² + 7x + 6
Then :
Δ = 7² - 4 × 2 × 6 = 1
and other roots of solutions are :
x1 = [tex]\frac{-7 + \sqrt{1} }{4}[/tex] = [tex]\frac{-3}{2}[/tex]
x2 = [tex]\frac{-7 - \sqrt{1} }{4}[/tex] = -2
So , the other factors will be x + [tex]\frac{3}{2}[/tex] and x + 2.
Therefore , the correct options are option A and option D that is the factors of given binomial expressions are x - 1 and x + 2.
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Given f(x)=2x^3+x^2-7x-6. Find all real and imaginary zeroes. Show your work.
The real zeroes of given function is [tex]-\frac{3}{2},-1, \text { and } 2[/tex]
Solution:Given that, [tex]f(x)=2 x^{3}+x^{2}-7 x-6[/tex]
We have to find the real and imaginary zeroes
This can be found out by equating the function to zero and finding the roots "x"
Now, let us use trail and error method.
So put x = 1 in f(x)
f(1) = 2 + 1 – 7 – 6 = - 10
1 is not a root. Since f(1) is not equal to 0
Now put x = -1
f(-1) = -2 + 1 + 7 – 6 = 0
-1 is a root. Since f(-1) is equal to 0
So, one of the roots is -1. Let the other roots be a, b.
[tex]\text { Sum of roots }=\frac{-x^{2} \text { coefficient }}{x^{3} \text { coefficient }}[/tex]
[tex]\begin{array}{l}{a+b+(-1)=\frac{-1}{2}} \\\\ {a+b=1-\frac{1}{2}} \\\\ {a+b=\frac{1}{2} \rightarrow(1)}\end{array}[/tex]
[tex]\begin{array}{l}{\text {Product of roots }=\frac{-\text {constant}}{x^{3} \text {coefficient}}} \\\\ {a b(-1)=\frac{-(-6)}{2}} \\\\ {a b(-1)=3} \\\\ {a b=-3 \rightarrow(2)}\end{array}[/tex]
Now, we know that, algebraic identity,
[tex]\begin{array}{l}{(a-b)^{2}=(a+b)^{2}-4 a b} \\\\ {(a-b)^2=\left(\frac{1}{2}\right)^{2}-4(-3)} \\\\ {(a-b)^2=\frac{1}{4}+12} \\\\ {(a-b)^2=\frac{49}{4}} \\\\ {a-b=\frac{7}{2} \rightarrow(3)}\end{array}[/tex]
Add (1) and (3)
[tex]\begin{array}{l}{2 a=\frac{7+1}{2} \rightarrow 2 a=4 \rightarrow a=2} \\\\ {\text { Then, from }(2) \rightarrow b=-\frac{3}{2}}\end{array}[/tex]
Hence, the roots of the given equation are [tex]-\frac{3}{2},-1, \text { and } 2[/tex]
There are 347 students at a college who have taken a course in calculus, 214 who have taken a course in discrete mathematics and 190 who have taken courses in both calculus and discrete mathematics.
A.How many students have taken a course in either calculus or discrete mathematics?
B.How many have taken calculus but not discrete mathematics?
C.How many have taken discrete mathematuics but not calculus?
Final answer:
There are 371 students who have taken a course in either calculus or discrete mathematics, 157 students have taken calculus but not discrete mathematics, and 24 students have taken discrete mathematics but not calculus.
Explanation:
To determine how many students have taken a course in either calculus or discrete mathematics, we use the principle of inclusion and exclusion. The formula for two sets A (calculus students) and B (discrete mathematics students) is: |A∪B| = |A| + |B| - |A∩B|.
A. Using the provided numbers: |A∪B| = 347 (calculus students) + 214 (discrete mathematics students) - 190 (students in both) = 371 students have taken a course in either calculus or discrete mathematics.
B. The number of students who have taken calculus but not discrete mathematics is found by subtracting the number of students who have taken both from the total number of calculus students: 347 - 190 = 157 students.
C. Similarly, the number of students who have taken discrete mathematics but not calculus is found by subtracting the number of students who have taken both from the total number of discrete mathematics students: 214 - 190 = 24 students.
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Keyshawn is bowling in a competition. He has a 26% chance of getting a strike each time he bowls. What is the probability that Keyshawn doesn't get a strike until after his first five attempts?
A) 0.19
B) 0.22
C) 0.78
D) 0.85
The probability of Coram making a free throw is 77%. What is the probability that he makes his first free throw within his first two shots?
A) 79%
B) 83%
C) 88%
D) 95%
He has a 26% of getting a strike, which means he has a 74% chance of not getting a strike ( 100% - 26% = 74%).
Multiply the chance of not getting a strike by the number of attempts:
0.74 x 0.74 x 0.74 x 0.74 x 0.74 = 0.22
The answer is B) 0.22
The probability of making a free throw is 77%, the probability of not making one would be 23% ( 100% - 77% = 23%).
Add the probability of making the first one ( 0.77) by the probability of making the second one multiplied by the probability of missing the second one ( 0.77x 0.23)
0.77 + (0.77 x 0.23)
0.77 + 0.18 = 0.95
The answer is D) 95%
Final answer:
To find the probability that Keyshawn doesn't get a strike until after his first five attempts, we need to find the probability of him not getting a strike on each individual attempt. The probability that Keyshawn doesn't get a strike until after his first five attempts is 0.19. To find the probability that Coram makes his first free throw within his first two shots, we need to find the probability of him making a free throw on the first or the second shot. The probability that Coram makes his first free throw within his first two shots is 95%.
Explanation:
To find the probability that Keyshawn doesn't get a strike until after his first five attempts, we need to find the probability of him not getting a strike on each individual attempt. Since each attempt is independent, we can multiply the probabilities together.
The probability of him not getting a strike on each attempt is 1 - 0.26 (the probability of getting a strike). So the probability of not getting a strike on the first five attempts is (1 - 0.26) * (1 - 0.26) * (1 - 0.26) * (1 - 0.26) * (1 - 0.26).
Calculating this expression, we get:
(0.74) * (0.74) * (0.74) * (0.74) * (0.74) = 0.192.
So the probability that Keyshawn doesn't get a strike until after his first five attempts is approximately 0.192.
Therefore, the correct answer is A) 0.19.
To find the probability that Coram makes his first free throw within his first two shots, we need to find the probability of him making a free throw on the first or the second shot. These events are mutually exclusive, so we can add the probabilities together.
The probability of him making a free throw on the first shot is 0.77, and the probability of him missing the first shot and making a free throw on the second shot is (1 - 0.77) * 0.77. So the total probability is 0.77 + (1 - 0.77) * 0.77.
Calculating this expression, we get:
0.77 + 0.23 * 0.77 = 0.77 + 0.1771 = 0.9471.
So the probability that Coram makes his first free throw within his first two shots is approximately 0.9471.
Therefore, the correct answer is D) 95%.
Find the formula for the nth term in this arithmetic sequence: 8,4,0,-4
A park ticket is $42 per person. For groups with up to 8 people, the cost per ticket goes down $3. Molly's ticket cost $30. How many people are in Molly's group?
There are 4 people in Molly's group.
Step-by-step explanation:
Let,
x be the number of people.
Individual ticket price = $42
Price after discount = $30
Discount per ticket = $3
Discount for x people = 3x
According to given statement;
42-3x=30
Subtracting 42 from both sides
[tex]42-42-3x=30-42\\-3x=-12\\[/tex]
Dividing both sides by -3
[tex]\frac{-3x}{-3}=\frac{-12}{-3}\\x=4[/tex]
There are 4 people in Molly's group.
Keywords: Subtraction, division
Learn more about subtraction at:
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Urban Encroachment is causing the area of a forest to decline at a rate of 7.62% per year . Use the exact half-life formula to determine the half-life of the forest .
A)7.50
B)8.75
C)17.50
D)1.08