The loan's future value will be $6925 when Lynne pays a simple interest rate of 5.4%.
According to the question,
We have the following information:
Lynne needs to borrow $6250 for cosmetic surgery. She obtains a loan from her grandmother for 24 months at a simple interest rate of 5.4%.
We know that the following formula is used to find the simple interest:
Principal*rate*time/100
Principal = $6250
Rate = 5.4%
Time = 2 years
Simple interest = (6250*5.4*2)/100
Simple interest = 67500/100
Simple interest = $675
Now, the total amount of loan will be:
Interest + principal
675+6250
$6925
Hence, the loan's future value will be $6925 when Lynne pays a simple interest rate of 5.4%.
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NO LINKS!! Please help me with these sequences Part 2x
Answer:
4. 121,800
5. 591 3/32
6. 1/4
Step-by-step explanation:
4. Sum of arithmetic sequenceYou want the sum of the first 200 terms of the arithmetic sequence starting 12, 18, 24, ....
The sum of the first n terms of an arithmetic sequence is given by the formula ...
Sn = (2·a1 +d(n -1))(n/2)
where a1 is the first term and d is the common difference. Your sequence has first term 12 and common difference 18-12 = 6. The desired sum is ...
S200 = (2·12 +6(200 -1))/(200/2) = (24 +1194)(100) = 121,800
5. Sum of geometric sequenceYou want the sum of the first 8 terms of the geometric sequence starting 12, 18, 27, ....
The sum of the first n terms of a geometric sequence is given by the formula ...
Sn = a1·(r^n -1)/(r -1)
where a1 is the first term and r is the common ratio. Your sequence has first term 12 and common ratio 18/12 = 3/2. The desired sum is ...
S8 = 12·((3/2)^8 -1)/(3/2 -1) = 24·6305/256 = 591 3/32
6. Sum of geometric sequenceFor this sequence, a1 = 1/12 and r = 2/3. When the sum is infinite and |r| < 1, the sum formula becomes ...
S = a1/(1 -r)
The desired sum is ...
S = (1/12)/(1 -2/3) = (1/12)/(4/12) = 1/4
Answer:
[tex]\textsf{4.} \quad 121,800[/tex]
[tex]\textsf{5.} \quad \dfrac{18915}{32}=591.09375[/tex]
[tex]\textsf{6.} \quad \dfrac{1}{4}[/tex]
Step-by-step explanation:
Question 4[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of an arithmetic series}\\\\$S_n=\dfrac{1}{2}n[2a+(n-1)d]$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $d$ is the common difference.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given arithmetic sequence:
12, 18, 24, ...The first term of the given sequence is 12.
The common difference can be found by subtracting the first term from the second term:
[tex]\implies d=a_2-a_1=18-12=6[/tex]
Therefore:
a = 12d = 6To find the sum of the first 200 terms, substitute n = 200, a = 12 and d = 6 into the formula:
[tex]\begin{aligned}S_{200}&=\dfrac{1}{2}(200)[2(12)+(200-1)(6)]\\&=100[24+(199)(6)]\\&=100[24+1194]\\&=100[1218]\\&=121800\end{aligned}[/tex]
Question 5[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=\dfrac{a(1-r^n)}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\ \phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given geometric sequence:
12, 18, 27, ...The first term of the given sequence is 12.
The common ratio can be found by dividing the second term by the first term:
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{18}{12}=1.5[/tex]
Therefore:
a = 12r = 1.5To find the sum of the first 8 terms, substitute n = 8, a = 12 and r = 1.5 into the formula:
[tex]\begin{aligned}\implies S_{8}&=\dfrac{12(1-1.5^8)}{1-1.5}\\\\&=\dfrac{12\left(1-\frac{6561}{256}\right)}{-0.5}\\\\&=\dfrac{12\left(-\frac{6305}{256}\right)}{-0.5}\\\\&=\dfrac{-\frac{18915}{64}}{-0.5}\\\\&=\dfrac{18915}{32}\\\\&=591.09375\end{aligned}[/tex]
Question 6[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Sum of an infinite geometric series}\\\\$S_{\infty}=\dfrac{a}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}[/tex]
Given geometric sequence:
[tex]\dfrac{1}{12},\dfrac{1}{18},\dfrac{1}{27},...[/tex]The first term of the given sequence is ¹/₁₂.
The common ratio can be found by dividing the second term by the first term:
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{\frac{1}{18}}{\frac{1}{12}}=\dfrac{2}{3}[/tex]
Therefore:
a = ¹/₁₂r = ²/₃To find the sum of the infinite geometric sequence, substitute a = ¹/₁₂ and r = ²/₃ into the formula:
[tex]\begin{aligned}\implies S_{\infty}&=\dfrac{\frac{1}{12}}{1-\frac{2}{3}}\\\\&=\dfrac{\frac{1}{12}}{\frac{1}{3}}\\\\&=\dfrac{1}{12} \times \dfrac{3}{1}\\\\&=\dfrac{3}{12}\\\\&=\dfrac{1}{4}\end{aligned}[/tex]
Levi has $0.74 worth of pennies and nickels. He has 2 more pennies than nickels.
Determine the number of pennies and the number of nickels that Levi has.
If he has 2 more pennies than nickels. the number of pennies is 10.6 and the number of nickels that Levi has is 12.6.
How to find the pennies and nickels?n = number of nickels
n-2 = number of pennies
Hence,
5n +1(n -2) = 74
Collect like terms
6n = 76
Divide both side by 6n
n = 76/6
n = 12.6 ( nickels)
Pennies
n-2
= 12.6 -2
= 10.6
Therefore Levi has 10.6 pennies and 12.6 nickels.
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Answer:
Step-by-step explanation:
Jb
One factor of f (x ) = 4 x cubed minus 4 x squared minus 16 x + 16 is (x – 2). What are all the roots of the function? Use the Remainder Theorem.
A factor of f(x) is (x – 1).
A factor of f(x) is (x + 1).
Both (x – 1) and (x + 1) are factors of f(x).
Neither (x – 1) nor (x + 1) is a factor of f(x).
The answer is option A) A factor of f(x) is (x – 1).
What is the remaining theorem?
Remainder Theorem is a method for dividing polynomials according to Euclidean geometry. This theorem states that when a polynomial P(x) is divided by a factor (x - a), which isn't really an element of the polynomial, a smaller polynomial is produced along with a remainder.
f(x) = 4x3 - 4x2 - 16x + 16 [Given]
One aspect is (x - 2)
The remaining theorem is used to:
Where q(x) is the quotient polynomial and r(x) is the remainder polynomial, f(x) = (x-2)q(x) + r(x).
As (x-2) is a factor of f, r(x) equals 0. (x)
f(x) = (x-2)·q(x) (x)
q(x) = f(x)/ (x-2) (x-2)
[4x3 - 4x2 - 16x + 16]/ (x - 2) (x - 2)
= [4x2 (x - 1) - 16 (x - 1)]
/ (x - 2) (x - 2)
By even more simplifying
= [(x - 1) (4x2 - 16)]
/ (x - 2) (x - 2)
excluding 4 as common
= [(x - 1) 4 (x2 - 4)]
/ (x - 2) (x - 2)
With the use of the algebraic formula a2 - b2 = (a + b) (a - b)
= [(x - 1) 4 (x + 2) (x - 2)]
/ (x - 2) (x - 2)
= [4 (x - 1) (x - 2) (x + 2)]
/ (x - 2) (x - 2)
The roots are therefore 2, -2, and 1.
there the ans is A) A factor of f(x) is (x – 1).
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Intercept and slope?
Answer:
Slope = -2
Y-intercept = -8
Step-by-step explanation:
Hello!
Slope-Intercept Form: y = mx + b
m = slopeb = y-interceptThe m variable in this equation would be -2, and it is also the slope of the equation.
The b variable in this equation is -8, which is the y-intercept of the equation.
Answer:
y intercept -8
slope -2
Step-by-step explanation:
y = -2x-8
This is written in slope intercept form
y = mx+b
The slope is m which is -2
The y intercept is b which is -8
A rectangular room, / m long and b m wide,
has a perimeter p, where p = 2l+2b.
a Find the perimeter of a room which is
3.5 m long and 2 m wide.
b Find the length of a room of perimeter
20 m and width 3 m.
The perimeter of the given rectangular room is found as -
Part a: p = 11 m.
Part b: p = 46 m.
Explain the term perimeter of rectangle?A rectangle's perimeter (P) is the sum of the lengths of its four sides.A rectangle has equal size lengths plus two equal widths since its opposite sides are equal.As, given in the question-
The rectangular room has-
l m long and b m wide,
perimeter p, where p = 2l+2b ...eq 1
Part a: perimeter of a room for, 3.5 m long and 2 m wide.
Put l = 3.5 and b = 2 in eq 1.
p = 2l+2b
p = 2(3.5) +2(2)
p = 7 + 4
p = 11 m.
Part b: perimeter of a room for, 20 m long and 3 m wide.
Put l = 20 and b = 3 in eq 1.
p = 2(20) +2(3)
p = 40 + 6
p = 46 m.
Thus, the perimeter of the given rectangular room is found as 46m.
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Select the correct answer. Which inequality can this number line represent? A. 7 − x ≥ 1 or 4x + 9 < 13 B. 3x + 2 < 5 or 2x − 1 ≥ 11 C. 7 − x ≥ 1 and 4x + 9 < 13 D. 3x + 2 < 5 and 2x − 1 ≥ 11
Answer: A
Step-by-step explanation: A
A
Please help me!!!! WILL GIVE BRAINLIEST!!!!
Answer: C & E
Step-by-step explanation: A is a lie, function 1 is pos
B is a lie function 1 is pos
C is true
D is a lie
E is true
A hospital would like to determine the mean length of stay for its patients having abdominal surgery. A sample of 17 patients revealed a sample mean of 6.6 days and a sample standard deviation of 1.5 days. Assume that the lengths of stay are approximately normally distributed. Find a 99% confidence interval for the mean length of stay for patients with abdominal surgery. Round the endpoints to two decimal places, if necessary.
The 99% confidence interval for the mean length of stay for patients with abdominal surgery is 5.54, 7.66
How to solve for confidence intervalWe have the following data to solve the question with
sample size = 17
standard deviation s = 1.5
Confidence interval = 99%
mean = 6.6 days
α = 1 - 0.99
= 0.01
df = degree of freedom = n -1
17 - 1 = 16
tα/2 = 0.01 / 2
= t 0.005, 16
t critical value = 2.9208
99 % confidence interval =
X ± t critical * s/√n
[tex]6.6 +-2.9208 * \frac{1.5}{\sqrt{17} }[/tex]
6.6 ± 1.0626
6.6 - 1.0626, 6.6 + 1.0626
= 5.54, 7.66
The 99% confidence interval for the mean length of stay for patients with abdominal surgery is 5.54, 7.66
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A concession stand at the Tennis Center sells a hamburger/drink combination dinner for $7. The profit, y (in dollars), can be approximated by
y=-0.001x² +
2+3.2x-400 where x is the number of dinners prepared.
(a) Find the number of dinners that should be prepared to maximize profit.
(h) What is the maximum profit?
The number of dinners that should be prepared to maximize profit is 1200.
The maximum profit is $1040
We know that,
y = ax²+ bx + c
The expression has the greatest value at x = -b/2a
From the question, we have
y=-0.001x² +2.4x-400
x = -b/2a
= -2.4/2*(-0.001)
=1200
substituting the value we get
y=-0.001x² +2.4x-400
y=-0.001*1200² +2.4*1200-400
y=$1040
Multiplication:
Finding the product of two or more numbers in mathematics is done by multiplying the numbers. It is one of the fundamental operations in mathematics that we perform on a daily basis. Multiplication tables are the main use that is obvious. In mathematics, the repeated addition of one number in relation to another is represented by the multiplication of two numbers. These figures can be fractions, integers, whole numbers, natural numbers, etc. When m is multiplied by n, either m is added to itself 'n' times or the other way around.
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Solve by factoring.
6^2 − − 12 = 0
Answer:
48
Step-by-step explanation:
1 ) Simplify...
[tex]6^2-\left(-12\right)[/tex]
[tex]=6^2+12[/tex]
[tex]6^2=36[/tex]
[tex]=36+12[/tex]
2 ) Add the numbers...
[tex]36+12 =[/tex]
[tex]48[/tex]
Hope this helps! :)
PLEASE HELPP! ORDER THE NUMBERS FROM LEAST TO GREATEST
Answer:
least
- 1/4 ft
- 14 3/4 ft
- 15 1/2 ft
- 20 3/5 ft
greatest
Answer:
- 20 3/5 ft
- 15 1/2 ft
- 14 3/4 ft
- 1/4 ft
I the number is larger looking but have a negative symbol, then it is smaller.
For example, -1 is greater than -6. Also -1/4 is greater than -20 3/5.
Trust me, I am a year ahead in school, so trust me.
I hope this helped.
What is the measure of angle ABC
Answer:
D
Step-by-step explanation:
[tex]m\angle EBD=33^{\circ} \implies m\angle EBC=33^{\circ} \\ \\ \therefore m\angle DBC=m\angle EBD+m\angle EBC=66^{\circ} \\ \\ m\angle DBC=66^{\circ} \implies m\angle ABD=66^{\circ} \\ \\ \therefore m\angle ABC=m\angle ABD+m\angle DBC=132^{\circ}[/tex]
Show that f(x)= 2x-3/2 & g(x)= 2x+3/2 are inverse fractions. Must show fog=x and gof=x. I really don’t understand this and it’s frustrating me. I must show work
Hence its proven that f and g are inverse of one another
What is a Function?
A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Given,
f(x)= 2x-3/2
g(x)= 2x+3/2
Step 1: show that fog=x
f(g(x))
[2((2x + 3) / 2) - 3]/2
[(2x + 3) - 3] / 2
(2x + 3 - 3) / 2
2x / 2
x , which is equals to x
Now, for Step 2: gof = x
g(f(x))
g((2x - 3)/2)
[2((2x - 3)/2) + 3]/2
[(2x - 3) + 3]/2
(2x - 3 + 3)/2
2x/2
x , which is equals to x
Since, both fog(x) and gof(x) equals to x
Hence, its proven that f and g are inverse of one another
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Jillian sells handmade t- Shirts and large purses; each tea shirt costs $15. Each Large purse goes for $59 and the cost of small purses is $32 each. In the first week she sold 4 t-shirts and 3 large purses and 9 small. In the second week she sold another 5 t-shirts, 7 large bags and 4 small ones. Last week she sold 8 t-Shirts and also sold 3 large purses and 5 small purses. What was the exact amount she made in 3 weeks?
Answer:
$1598
Step-by-step explanation:
1st week
4*15=60
32*9=288
59*3=177
60+177+288=525
2nd week
5*15=75
59*7=413
32*4=128
75+413+128=616
3rd week
8*15=120
3*59=177
5*32=160
120+177+160=457
525+616+457=$1598
1x(3x6)=18 create a equivalent multiplication sentence that illustrates the associative property of multiplication
The equivalent multiplication sentence that shows associative property is 3x(1×6)
What is associative property?Associative property is defined as, when more than two numbers are added or multiplied, the result remains the same, irrespective of how they are grouped.
This means that if a,b,c are real numbers then a×(b×c)=b×(a×c)
using the associative property,1×(3×6)= 3×(1×6)
To verify the statement,let's find out
solving the parentheses first
1×(3×6)=1×18= 18
3×(1×6)= 3×6= 18
this means that 1×(3×6) and 3×(1×6) are associative
therefore 1×(3×6)=3×(1×6)
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help please! Trig question on homework this is the only one I can’t really figure out!
The measure of central angle is 45°
Here, arc length S = 5π/4
radius r = 5
We need to find central angle θ
Using the formula of arc length,
S = r * θ
5π/4 = 5 * θ
θ = π/4
θ = 45°
Therefore, the measure of central angle is 45°
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Beth makes $10.25/hour and Bartley makes $11.50 an hour. If Beth worked for 23 hours and Bartley worked for 17 hours, what was their difference in the amount of money they each earned?
The difference in the amount of money earned by Beth and Bartley is $40.25.
Given that, Beth makes $10.25/hour and Bartley makes $11.50 an hour.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
If Beth worked for 23 hours
Money earned by Beth is
10.25×23=$235.75
If Bartley worked for 17 hours
Money earned by Bartley is
11.50×17 =$195.5
So, difference of money = 235.75-195.5
= $40.25
Hence, the difference in the amount of money earned by Beth and Bartley is $40.25.
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Identify the rate of change in each equation.
Answer:
7.
[tex] \frac{1}{4} [/tex]
8.
[tex]2[/tex]
9.
[tex] - \frac {5}{4} [/tex]
10.
[tex]3[/tex]
Step-by-step explanation:
rate of change is slop
(7/4x - 5) - (2y - 3.5) - (-1/4x + 5)
The value of the expression given as (7/4x - 5) - (2y - 3.5) - (-1/4x + 5) is 2x - 2y - 6.5
How to determine the solution of the expression?In this question, the representation of the expression is given as
(7/4x - 5) - (2y - 3.5) - (-1/4x + 5)
Start by removing the brackets in the expression
So, we have
7/4x - 5 - 2y + 3.5 + 1/4x - 5
Evaluate the fractions in the expression
The expression becomes
1.75x - 5 - 2y + 3.5 + 0.25x - 5
Collect the like terms
1.75x + 0.25x - 2y - 5 + 3.5 - 5
Evaluate the like terms
2x - 2y - 6.5
Hence, the solution is 2x - 2y - 6.5
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Make the subject of
6x = t
The subject of the equation is x and it can be written as x = t/6.
What do you mean by solving an equation?Calculating the value of the unknown variable while keeping the equation balanced on both sides is the process of solving equations. The equation's solution is the value of the variable for which the equation is true. An equation remains the same even when the LHS and RHS are flipped. The solution is discovered after isolating the variable for which the value must be determined. Depending on the kind of equation we are dealing with, we can solve it. There are various types of equations, including linear, quadratic, rational, and radical ones.
The given equation is
6x = t
Divide both sides by 6
6x/ 6 = t/ 6
x = t/ 6
So, the subject of the equation is x - t/6.
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what is -3/4x - 1/3 + 7/8x - 1/2 written in siplest form?
help me pls
Answer: 1/8x - 5/6
Step-by-step explanation:
Let's move the similar terms of the problem together.
So first we have:
-3/4x + 7/8x
And then we have:
-1/3 - 1/2
So to add/subtract fractions, the denominators have to be equal. We can change the -3/4 into -6/8 by just multiplying by 2/2 which is just 1. (Anything multiplied by 1 is just itself). So then we have:
-6/8x + 7/8x ---> 1/8x
Next, the second part.
Since neither denominators (3 or 2) can turn into one another by multiplying them with a number, we'll just multiply them with the opposite denominator.
So then we have:
-1/3 * (2/2) = -2/6
-1/2 * (3/3) = -3/6
-2/6 - 3/6 = -5/6
So finally we have:
1/8x - 5/6
A Mrs. Sharpe’s Starbucks card has a balance of $37.45 remaining. If each visit she spends$5.35, which equation represents the number of visits, v, she may make using the card?
Answer:
the equation is 37.45-5.35v=0
Step-by-step explanation:
37.45-5.35v=0
minus 37.45 to both sides
-5.35v=-37.45
divide -5.35 from both sides
v=7
after 7 days she will finish the card
K
A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 59 tablets, then accept the whole batch if there is
only one or none that doesn't meet the required specifications. If one shipment of 3000 aspirin tablets actually has a 2% rate of defects, what is the probability that this whole
shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?
The probability that this whole shipment will be accepted is
(Round to four decimal places as needed.)
4
Answer:
The probability that this whole shipment will be accepted is 72.39%.Step-by-step explanation:
The probability of no defects with 2% rate of defects:
P(no defects) = 59*0.02⁰*(1 - 0.02)⁵⁹ = 0.3583 (rounded) = 35.83%Probability of exactly one defect:
P(1 defect) = 59*0.02¹*(1 - 0.02)⁵⁸ = 0.3656 (rounded) = 36.56%Probability of one or no defects:
P(1 or none) = 35.83% + 36.56% = 72.39%This indicates that:
100% - 72.39% = 27.61% of shipments will be rejected, it is a lot, so many shipments will be rejectedThe probability that this whole shipment will be accepted is 72.39%.
What is the graphing form of y = x2 - 12x + 7
The vertex of the equation y=x^2-12x+7 is (6,-29) and the graph is given below.
In the given question we have to find the graphing form of y=x^2-12x+7.
The given equation is y=x^2-12x+7.
To graph the given equation we firstly express that equation in the standard form of parabola.
y=a(x-h)^2+k
Add and subtract 36 in the given equation;
y=x^2-12x+36-36+7
y=x^2-2×6×x+(6)^2-29
y=(x-6)^2-29
The vertex of the given equation is (6,-29).
The graph of the given equation is given below:
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A 17 inch candle burns down in 10 hours. At what rate does the candle burn, in inches per hour?
Answer:
1.7
Step-by-step explanation:
[tex]\frac{17 \text{ in}}{10 \text{ hr}}=1.7[/tex] in/hr
The function f(x) is graphed below. What is true about the graph on the interval from point a to point b?
Answer:
Option 1. It is positive and increasing.
Step-by-step explanation:
In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers (1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. The cards in Ms. Q's deck are shown below.
Yunseol draws 5 cards from Ms. Q's deck. What is the probability that three cards have the same number?
The probability that three cards have the same number, using the hypergeometric distribution, is of:
0.0448 = 4.48%.
What is the hypergeometric distribution formula?The probability mass function for the hypergeometric distribution is defined as follows:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters of the function are listed below:
x is the number of successes in the sample.N is the size of the population.n is the size of the sample.k is the total number of desired outcomes in the population.In the context of this problem, the values of these parameters are given as follows:
N = 28, k = 4, n = 5, x = 3.
Hence for one color, the probability that three cards have the same number is:
[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]
[tex]P(X = 3) = h(3,28,5,4) = \frac{C_{4,3}C_{24,2}}{C_{28,5}} = 0.0112[/tex]
There are four colors, hence the probability is:
p = 4 x 0.0112 = 0.0448 = 4.48%.
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Draw the image of AABC under the translation (x, y) → (x, y + 3).
What is the equation of a line that is perpendicular to y=0.25x−7 and passes through the point (-6,8)
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above
[tex]0.\underline{25}\implies \cfrac{25}{1\underline{00}}\implies \cfrac{1}{4} \\\\[-0.35em] ~\dotfill\\\\ y=0.25x-7\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{4}}x-7\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{\cfrac{1}{4}} ~\hfill \stackrel{reciprocal}{\cfrac{4}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{4}{1}\implies -4}}[/tex]
so we're really looking for the equation of a line whose slope is -4 and it passes through (-6 , 8)
[tex](\stackrel{x_1}{-6}~,~\stackrel{y_1}{8})\hspace{10em} \stackrel{slope}{m} ~=~ - 4 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{8}=\stackrel{m}{- 4}(x-\stackrel{x_1}{(-6)}) \implies y -8= -4 (x +6) \\\\\\ y-8=-4x-24\implies {\Large \begin{array}{llll} y=-4x-16 \end{array}}[/tex]
The vertices of a rectangle are located at the coordinates (1, 4), (1, 5), (6, 5), and (6, 4). Find the length of the sides of the rectangle and its perimeter.
Answer: This is a 1 by 5 rectangle
perimeter = 12 units
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Work Shown:
Label the vertices A,B,C,D
A = (1, 4)
B = (1, 5)
C = (6, 5)
D = (6, 4)
The x coordinates of points A and B are the same (x = 1). The difference in y values of these points is 5-4 = 1 unit. This is the height of the rectangle. In other words, it's the length of segment AB.
Points B and C have the same y coordinate (y = 5). The difference in x value is 6-1 = 5 which is the horizontal length of the rectangle. This is the length of segment BC.
In short, this is a 1 by 5 rectangle. It's one unit tall and five units across.
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Perimeter = 2*(length+width)
Perimeter = 2*(5+1)
Perimeter = 2*6
Perimeter = 12 units
Or you can add up the four sides (1,5,1 and 5) to get the same result.