Answer:
y = (2x + 5)(2x + 3)
Step-by-step explanation:
y = 4x² + 16x + 15
consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term
product = 4 × 15 = 60 and sum = + 16
the factors are + 10 and + 6
use these factors to split the x- term
y = 4x² + 10x + 6x + 15 ( factor the first/second and third/fourth terms )
y = 2x(2x + 5) + 3(2x + 5) ← factor out (2x + 5) from each term
y = (2x + 5)(2x + 3)
since multiplication is commutative , then could be
y = (2x + 3)(2x + 5)
Celeste is planting a rectangular flower garden in which the width will be 4 feet less than its length. She has decided to put a birdbath within the garden that will occupy a space 3feet by 4 feet how many feet are now left for planting? Express your answer on factored form
Answer:
(L-6)(L+2)
Step-by-step explanation:
Let L be the length of the flower garden.
Then the width will be L-4.
The area of the flower garden = L*(L-4) =L²-4L
The area of the birdbath is 3*4 = 12 ft²
The area of the remaining space for planting is
= Area of flower garden - area of birdbath
L² - 4L - 12We can factor the expression as follows:
L² - 4L - 12 L²-(6-2)L-12L²-6x+2x-12taking common frome each two terms
L(L-6)+2(L-6)(L-6)(L+2)Therefore, the number of feet left for planting is (L-6)(L+2) in factored form.
The amount of time that people spend at Grover Hot Springs is normally distributed with a mean of 68 minutes and a standard deviation of 14 minutes. Suppose one person at the hot springs is randomly chosen. Let X = the amount of time that person spent at Grover Hot Springs . Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(
68
Correct,
14
Correct)
b. Find the probability that a randomly selected person at the hot springs stays longer then 81 minutes.
c. The park service is considering offering a discount for the 8% of their patrons who spend the least time at the hot springs. What is the longest amount of time a patron can spend at the hot springs and still receive the discount?
minutes.
d. Find the Inter Quartile Range (IQR) for time spent at the hot springs.
Q1:
minutes
Q3:
minutes
IQR:
minutes
a. The distribution of X is X ~ N(68, 14).
b. The corresponding area to the right of 0.9286, which is approximately 0.1772.
c. The longest amount of time a patron can spend and still receive the discount is approximately 48.5654 minutes.
d. The Inter Quartile Range (IQR) for time spent at the hot springs is approximately 21.373 minutes.
a. The distribution of X is X ~ N(68, 14), where X represents the amount of time a person spends at Grover Hot Springs, 68 is the mean, and 14 is the standard deviation.
b. To find the probability that a randomly selected person stays longer than 81 minutes, we need to calculate the area under the normal curve to the right of 81.
Using the z-score formula: z = (x - μ) / σ, where x is the value (81), μ is the mean (68), and σ is the standard deviation (14).
Plugging in the values, we have z = (81 - 68) / 14 = 0.9286.
Using a standard normal distribution table or a calculator, we can find the corresponding area to the right of 0.9286, which is approximately 0.1772.
c. To find the longest amount of time a patron can spend at the hot springs and still receive the discount, we need to find the value that corresponds to the lowest 8% of the distribution.
Using a standard normal distribution table or a calculator, we can find the z-score that corresponds to the 8th percentile, which is approximately -1.4051.
Using the z-score formula, we can calculate the longest amount of time: x = μ + z [tex]\times[/tex] σ = 68 + (-1.4051) [tex]\times[/tex] 14 = 48.5654 minutes.
Therefore, the longest amount of time a patron can spend and still receive the discount is approximately 48.5654 minutes.
d. The Inter Quartile Range (IQR) is a measure of the spread of the data and represents the range between the first quartile (Q1) and the third quartile (Q3).
To find Q1 and Q3, we can use the z-score formula and the standard normal distribution table.
For Q1, we find the z-score corresponding to the 25th percentile, which is approximately -0.6745.
Using the formula Q1 = μ + z [tex]\times[/tex] σ, we have Q1 = 68 + (-0.6745) [tex]\times[/tex] 14 = 57.053.
Therefore, Q1 is approximately 57.053 minutes.
For Q3, we find the z-score corresponding to the 75th percentile, which is approximately 0.6745.
Using the formula Q3 = μ + z [tex]\times[/tex] σ, we have Q3 = 68 + (0.6745) [tex]\times[/tex] 14 = 78.426.
Therefore, Q3 is approximately 78.426 minutes.
Finally, we can calculate the IQR by subtracting Q1 from Q3: IQR = Q3 - Q1 = 78.426 - 57.053 = 21.373 minutes.
Therefore, the Inter Quartile Range (IQR) for time spent at the hot springs is approximately 21.373 minutes.
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Select the correct answer. If function g has the factors (x − 7) and (x + 6), what are the zeros of function g? A. -7 and 6 B. -6 and 7 C. 6 and 7 D. -7 and -6
Answer:
-6 and 7.
Step-by-step explanation:
If we have a function called g, and we know that it has two factors: (x - 7) and (x + 6), then we can find the values of x that make g equal to zero. We call those values the "zeros" of the function g. To find the zeros, we just need to solve the equation (x - 7)(x + 6) = 0. The answer is that the zeros of g are -6 and 7.
Which of the following steps indicates the addition property of equality while solving the equation –1 – 6x = x – 15?
A) x = 14∕2
B) –1 – 6x = x – 15
C) 23 – 6x – 24 = x – 15
D) –1 – 6x + 15 = x – 15 + 15
Answer:
-1 - 6x = x - 15
Add 15 to both sides using the addition property of equality.
14 - 6x = x
14 = 7x
2 = x
D) -1 - 6x + 15 = x - 15 + 15
Answer and Step-by-step explanation:
Please refer to the photo for the solution!
Find the slope and the y-intercept of the following linear equation. 5. 3x + 2y = 14
Answer:
slope = - [tex]\frac{3}{2}[/tex] , y- intercept = 7
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given
3x + 2y = 14 ( subtract 3x from both sides )
2y = - 3x + 14 ( divide through by 2 )
y = - [tex]\frac{3}{2}[/tex] x + 7 ← in slope- intercept form
with slope m = - [tex]\frac{3}{2}[/tex] and y- intercept c = 7
0.059 and 0.01 which is greater?
Are the experimental probabilities after 300 trials closer to the theoretical probabilities?
After 300 trials, the experimental probabilities may not align perfectly with the theoretical probabilities. However, with more trials, the experimental probabilities tend to converge towards the theoretical probabilities for closer alignment.
To examine whether experimental probabilities after 300 trials align closely with theoretical probabilities, let's consider an example of flipping a fair coin.
Theoretical probability: When flipping a fair coin, the theoretical probability of obtaining heads or tails is 0.5 each. This assumes that the coin is unbiased and has an equal chance of landing on either side.
Experimental probability: After conducting 300 trials of flipping the coin, we record the outcomes and calculate the experimental probabilities. Let's assume that heads occurred 160 times and tails occurred 140 times.
Experimental probability of heads: 160/300 = 0.5333
Experimental probability of tails: 140/300 = 0.4667
Comparing the experimental probabilities to the theoretical probabilities, we can observe that the experimental probability of heads is slightly higher than the theoretical probability, while the experimental probability of tails is slightly lower.
In this particular example, the experimental probabilities after 300 trials do not align perfectly with the theoretical probabilities. However, it is important to note that these differences can be attributed to sampling variability, as the experimental outcomes are subject to random fluctuations.
To draw a more definitive conclusion about the alignment between experimental and theoretical probabilities, a larger number of trials would need to be conducted. As the number of trials increases, the experimental probabilities tend to converge towards the theoretical probabilities, providing a closer alignment between the two.
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The question probable may be:
Do experimental probabilities after 300 trials tend to align closely with theoretical probabilities? Consider an example scenario and calculate both the theoretical and experimental probabilities to determine if they are close.
Toula owns the Pita Pan restaurant. She needs to order supplies for the upcoming weekend rush. She needs 150 bags of pita bread. The bread come in crates of 50, and each crate costs $15.00. She also needs 65 containers of hummus dip. There are 5 containers in a box, and each box costs $20.00 What expressions can Toula use to determine how much the pita bread and hummus dips will cost? What will the total be?
The total cost of the pita bread and hummus dips will be $305.00.
To determine the cost of the pita bread and hummus dips, Toula can use the following expressions:
Cost of pita bread:
Number of crates needed = (150 bags) / (50 bags/crate) = 3 crates
Cost of each crate = $15.00
Total cost of pita bread = (Number of crates needed) × (Cost of each crate) = 3 crates × $15.00/crate = $45.00
Cost of hummus dips:
Number of boxes needed = (65 containers) / (5 containers/box) = 13 boxes
Cost of each box = $20.00
Total cost of hummus dips = (Number of boxes needed) × (Cost of each box) = 13 boxes × $20.00/box = $260.00
Therefore, the expressions Toula can use to determine the costs are:
Cost of pita bread = 3 crates × $15.00/crate
Cost of hummus dips = 13 boxes × $20.00/box
The total cost will be the sum of the costs of pita bread and hummus dips:
Total cost = Cost of pita bread + Cost of hummus dips
Total cost = $45.00 + $260.00
Total cost = $305.00
Therefore, the total cost of the pita bread and hummus dips will be $305.00.
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Shawn wants to paint all the surfaces of the table shown below.
A. the volume of 3 rectangular prisms
B. the surface area of 1 triangle and 4 cylinders
C. the volume of 1 rectangular prism and 3 cylinders
D. the surface area of 2 triangles and 1 rectangular prism
What's the answer? How do I solve for this?!
the answer is D
The figure can be divided into a rectangle and 2 triangles
Find the linear function
The linear function for this case is:
f(x) = 5,000*x + 7,000
How to find the linear function?The general linear function is written as:
f(x) = a*x + b
Where a is the slope and b is the y-intercept.
Here we want a linear function for the given scenario, we know that the initial population is 7,000, then we can write:
f(x)= a*x + 7,000
Then we know that the population increases by 5,000 per year for 5 years, so the slope is 5,000, then we can write the function as:
f(x) = 5,000*x + 7,000
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Which linear equation shows a proportional relationship?
y equals negative one sixth times x
y equals one sixth times x minus 8
y = −6x + 1
y = 6
Answer:
y = (-1/6)x represents a proportional relationship.
Sadie is going to see a movie and is taking her 4 kids. Each movie ticket costs $15 and there are an assortment of snacks available to purchase for $4.50 each. How much total money would Sadie have to pay for her family if she were to buy 6 snacks for everybody to share? How much would Sadie have to pay if she bought x x snacks for everybody to share?
Answer: 102 dollars/4.5x+75
Step-by-step explanation: Your question isn't really straightforward, x x snacks for everybody to share? Please elaborate, and are they talking about the total cost, or just the cost of 6 snacks?
First, we have to take into account that if SADIE is taking her FOUR kids, there will be 5 people.
Cost of tickets is equal to $15 per one, and
5x (where x=15, per 5)
5(15)=75, and now onto the snacks
4.5x2=9, and 6/2=3, so 9x3=27, or 4.5x6=27
75+27=102
then for x snacks, if 1 snack costs 4.5 dollars than it'd be 4.5x (x number of snacks)+75 to find the total cost with tickets and all.
A gaming system costs $600 and is on sale for 15% off. After the discount, there is a 5% tax. What is the final price of the gaming system?
Answer$535.50
Step-by-step explanation:
15% is equal to .15
So, multiply 600.00x .15=90
600.00 - 90.0=510.
510. 00x .05=25.50
510.00+25.50=535.50
Your answer is $535.5
Triangle NMO has vertices at N(−5, 2), M(−2, 1), and O(−3 , 3). Determine the vertices of image N′M′O′ if the preimage is reflected over x = −1.
N′(5, −2), M′(2, 1), O′(3, 3)
N′(−5, 5), M′(−2, 3), O′(−3, 7)
N′(3, 2), M′(0, 1), O′(1, 3)
N′(−5, −2), M′(−2, −1), O′(−3, −3)
The vertices of the image triangle N'M'O' are N'(5, 2), M'(2, 1), and O'(3, 3).
To determine the vertices of the image N'M'O' after reflecting triangle NMO over the line x = -1, we need to apply the reflection transformation to each vertex.
For a reflection over the line x = -1, we can find the image of a point (x, y) by finding its reflection as (2(-1) - x, y).
Applying this transformation to each vertex of triangle NMO, we get:
N' = (2(-1) - (-5), 2) = (5, 2)
M' = (2(-1) - (-2), 1) = (2, 1)
O' = (2(-1) - (-3), 3) = (3, 3)
The vertices of the image triangle N'M'O' are N'(5, 2), M'(2, 1), and O'(3, 3).
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Identify which year’s population will hit zero
The year when the population becomes zero is 2020.2
Which year the population will reach zero?We know that the population decreases at a constant rate, so it is modeled by a line:
y = ax + b
Where a is the slope and b is the y-intercept.
Here we have two points (2010, 5100) and (2012, 4100)
The slope is the quotient between the differences of the y-values and the x-values, then:
a = (4100 - 5100)/(2012 - 2010) = -1000/2 = -500
We can write.
y = -500*x + b
We know that when x = 2010, y = 5100, replacing that we get:
5100 = -500*2010 + b
b = 5100 + 500*2010 = 1,010,100
Then:
y = -500*x + 1,010,100
And it is zero when:
0 = -500*x + 1,010,100
x = 1,010,100/500
x = 2020.2.
That is the year when the pópulation becomes zero.
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quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a standard deviation of 62
minutes with a mean life of 606
minutes.
If the claim is true, in a sample of 99
batteries, what is the probability that the mean battery life would be greater than 619
minutes? Round your answer to four decimal places.
Answer:
Step-by-step explanation:
x = 13, ¿cuál ecuación es verdadera?
3(18 - x) = 67
4(9x) = 23
2(x-3)=7
5(x-9) = 20
When x = 13, the equation that is true is option D) 5(x - 9) = 20.
To determine which equation is true when x = 13, we can substitute the value of x into each equation and see which equation holds true. Let's go through each option:
A) 3(18 - x) = 67
Substituting x = 13:
3(18 - 13) = 67
3(5) = 67
15 = 67
The equation is not true when x = 13. Therefore, option A is false.
B) 4(9x) = 23
Substituting x = 13:
4(9*13) = 23
4(117) = 23
468 = 23
Again, the equation is not true when x = 13. Therefore, option B is also false.
C) 2(x - 3) = 7
Substituting x = 13:
2(13 - 3) = 7
2(10) = 7
20 = 7
Once again, the equation is not true when x = 13. Therefore, option C is false as well.
D) 5(x - 9) = 20
Substituting x = 13:
5(13 - 9) = 20
5(4) = 20
20 = 20
Finally, the equation is true when x = 13. Therefore, option D is true.
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Note: the translated questions is
X = 13, which equation is true?
Five clubs at Johnson School raised $2000. The incomplete circle graph shows what percent of the money was raised by each club. How much money did the Math Club raise?
$500
$600
$200
$400
$300
The Math Club raised $400.
To find out how much money the Math Club raised, we need to determine the percentage of the total amount raised that corresponds to the Math Club's portion.
Let's assume the Math Club raised "x" amount of money. The total amount raised by all five clubs is $2000.
According to the incomplete circle graph, the Math Club's percentage is missing, but we know the percentages for the other clubs: Computer Club raised 15%, Gardening Club raised 18%, Art Club raised 30%, and Spanish Club raised 17%.
To find the missing percentage for the Math Club, we subtract the percentages of the other clubs from 100%:
Missing percentage = 100% - (15% + 18% + 30% + 17%) = 100% - 80% = 20%
Now we can set up a proportion to determine the amount raised by the Math Club:
(x / $2000) = 20% / 100%
Cross-multiplying:
x = ($2000 * 20%) / 100%
Simplifying:
x = $400
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A merchant mixed 12 lb of a cinnamon tea with 5 lb of spice tea. The 17-pound mixture cost $28. A second mixture included 14 lb of the cinnamon tea and 6 lb of the spice tea. The 20-pound mixture cost $33. Find the cost per pound of the cinnamon tea and of the spice tea.
Cinnamon tea costs $1.50 per pound, and spice tea costs $2.75 per pound.
To solve this problem, we can set up a system of equations based on the given information.
Let's denote the cost per pound of the cinnamon tea as C, and the cost per pound of the spice tea as S.
From the first mixture, we know that the total weight is 17 pounds, so we can write the equation:
12C + 5S = 28 ----(Equation 1)
From the second mixture, we know that the total weight is 20 pounds, so we can write the equation:
14C + 6S = 33 ----(Equation 2)
To solve this system of equations, we can use a method like substitution or elimination.
Let's use the elimination method to eliminate the variable C:
Multiply Equation 1 by 2 and Equation 2 by -3 to eliminate the C terms:
24C + 10S = 56 ----(Equation 3)
-42C - 18S = -99 ----(Equation 4)
Add Equation 3 and Equation 4:
-18C - 8S = -43
Solve for S:
8S = 43 - 18C
S = (43 - 18C)/8 ----(Equation 5)
Now substitute Equation 5 into Equation 1:
12C + 5((43 - 18C)/8) = 28
Multiply through by 8 to eliminate the fraction:
96C + 215 - 90C = 224
6C = 9
C = 9/6 = 1.5
Substitute the value of C back into Equation 5 to find S:
S = (43 - 18(1.5))/8 = 2.75
Therefore, the cost per pound of the cinnamon tea is $1.50, and the cost per pound of the spice tea is $2.75.
In summary, the cost per pound of the cinnamon tea is $1.50, and the cost per pound of the spice tea is $2.75.
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you ran 4 1/2 times around a quarter mile track. how far did you run?
Answer:
1 1/8 of a mile.
Step-by-step explanation:
The distance around the track is one quarter of a mile. Therefore, if you run around the track 4 times, you will have ran 1 mile, as 4 * 1/4 = 1. You would also run the other 1/2 of the lap, and to find that distance, you would multiply 1/2 * 1/4, because you only ran 1/2 of a lap and not one whole lap, which would come out to 1/8 of a mile. So, your final answer would be 1 + 1/8 of a mile, which comes out to 1 and 1/8 of a mile.
D = {x|x is a whole number} E = {x|x is a perfect square between 1 and 9} F = {x|x is an even number greater than or equal to 2 and less than 9} Which of the following is an element of D ∩ (E ∩ F)? 16 3 6 4
The element 4 is an element of D ∩ (E ∩ F).
To find the intersection of sets D, E, and F, we need to determine the elements that are common to all three sets.
Set D consists of all whole numbers, so any whole number can be an element of set D.
Set E consists of perfect squares between 1 and 9. The perfect squares in this range are 1, 4, and 9.
Set F consists of even numbers greater than or equal to 2 and less than 9.
The even numbers in this range are 2, 4, 6, and 8.
Taking the intersection of sets E and F, we find that the common element is 4, as it is the only number that satisfies both conditions of being a perfect square and an even number in the given range.
Finally, taking the intersection of set D with the intersection of sets E and F, we find that the element 4 is also an element of set D ∩ (E ∩ F).
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se logarithms to solve the problem.
The rule of 70 is a rule of thumb for estimating the doubling time of a quantity (e.g., investment, GDP, population) experiencing growth that is compounded continuously. The rule states that if the growth rate is r% per year, then the time it takes for the quantity to double is approximately 70/r years.
(a)
Use the rule of 70 to estimate the time it takes for an investment to double in value if it grows at the rate of 5% per year compounded continuously.
yr
(b)
What is the exact time it will take for the investment in part (a) to double in value? (Round your answer to two decimal places.)
yr
a. The investment to double in value take about 14 years for the funding to double in value.
b. The genuine time it will take for the funding to double in fee is about 13.86 years.
(a) To estimate the time it takes for an funding to double in cost the use of the rule of 70, we want to decide the increase rate. In this case, the increase price is given as 5% per 12 months compounded continuously.
Using the rule of 70, we can calculate the estimated doubling time:
Time to double ≈ 70 / boom rate
Time to double ≈ 70 / 5
Simplifying, we have:
Time to double ≈ 14 years
Therefore, it would take about 14 years for the funding to double in value.
(b) To decide the genuine time it will take for the funding to double in value, we can use the formulation for non-stop compounding:
Doubling time (exact) = ln(2) / (ln(1 + r))
where r is the increase fee as a decimal.
In this case, the increase charge is 5% per year, or 0.05 as a decimal.
Doubling time (exact) = ln(2) / (ln(1 + 0.05))
Doubling time (exact) ≈ 13.86 years (rounded to two decimal places)
Therefore, the genuine time it will take for the funding to double in fee is about 13.86 years.
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The sum of three numbers is 71. The third number is 2 times the first. The second number is 5 less than the first. What are the numbers?
Answer:
19, 14, 38
Step-by-step explanation:
Let x, y, and z be each number respectively:
[tex]x+y+z=71\\z=2x\\y=x-5\\\\x+y+z=71\\x+(x-5)+2x=71\\2x-5+2x=71\\4x-5=71\\4x=76\\x=19\\\\y=x-5\\y=19-5\\y=14\\\\z=2x\\z=2(19)\\z=38[/tex]
Therefore, the three numbers are 19, 14, and 38.
What is the solution to x – 5 + 2 < 20? –7 < x < 15 –13 < x < 23 x < –7 or x > 15 x < –13 or x > 23
Answer:
Therefore, the correct answer is: x < 23.
Step-by-step explanation:
To solve the inequality x - 5 + 2 < 20, we can simplify it step by step:
x - 5 + 2 < 20
Combine like terms:
x - 3 < 20
Add 3 to both sides of the inequality:
x - 3 + 3 < 20 + 3
Simplify:
x < 23
The solution to the inequality is x < 23.
Therefore, the correct answer is: x < 23.
Answer and Step-by-step explanation:
Please see the photo for the solution :)
Please look at photo. I’ll give good rating!
An output value for (fog)(x) is 55/(x² + 2x).
Domain = (-∞, 1) U (-2, 0) U (0, ∞) or {x|x ≠ 0, -2}.
How to determine the corresponding composite function?In this exercise, we would determine the corresponding composite function of f(x) and g(x) under the given mathematical operations in simplified form as follows;
(fog)(x) = 5/(x + 2) × 11/x
(fog)(x) = 55/x(x + 2)
(fog)(x) = 55/(x² + 2x)
For the restrictions on the domain, we would have to equate the denominator of the rational function to zero and then evaluate as follows;
x² + 2x ≠ 0
x² ≠ -2x
x ≠ -2
Domain = (-∞, 1) U (-2, 0) U (0, ∞) or {x|x ≠ 0, -2}.
In conclusion, we can reasonably infer and logically deduce that x must not be equal to 0 and -2.
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Find the equation of the line in slope-intercept form, parallel to a line joining the points (1,-2) and (-4,3) and passing through (-4,-5).
I
The equation of the line parallel to a line joining points (1,-2) and (-4,3) and passing through (-4,-5) is
(Simplify your answer. Type your answer in slope-intercept form.)
The equation of the line parallel to the line passing through (1, -2) and (-4, 3) and passing through the point (-4, -5) is y = -x - 9 in slope-intercept form.
To find the equation of a line parallel to a given line, we need to determine the slope of the given line and then use it to construct the equation of the parallel line.
First, let's calculate the slope of the given line passing through points (1, -2) and (-4, 3). The slope, denoted as m, can be found using the slope formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates, we have:
m = (3 - (-2)) / (-4 - 1) = 5 / (-5) = -1
Now that we have the slope, we can use it to construct the equation of the parallel line.
We'll use the point-slope form of a linear equation, which is:
y - y1 = m(x - x1)
where (x1, y1) represents the coordinates of a point on the line.
We'll use the point (-4, -5) on the parallel line:
y - (-5) = -1(x - (-4))
y + 5 = -1(x + 4)
Simplifying further:
y + 5 = -x - 4
y = -x - 9
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Given rhombus QRST, find the
perimeter if QU = 3 and RU equals 4.
Q
R
T
U
X
S
The perimeter of the rhombus in this problem is given as follows:
19.8 units.
What is the perimeter of a polygon?The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.
The diagonal length can be obtained as follows:
QU = US = 3.RU = UT = 4.RU + UT = 7.
Applying the Pythagorean Theorem, the side length is obtained as follows:
x² + x² = 7²
2x² = 49
[tex]x = \sqrt{\frac{49}{2}}[/tex]
x = 4.95.
Then the perimeter is given as follows:
P = 4 x 4.95
P = 19.8 units.
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Arc BC on circle A has a length of 115,
- inches. What is the radius of the circle?
115/6 pi
138°
The radius of the circle is 25 inches. The length of arc with a central angle of 138° is 115π/6 in
What is an equation?An equation is an expression that shows how numbers and variables are related to each other using mathematical operators.
The length of an arc with a central angle Ф with circle radius (r) is given by:
Length of arc = (Ф/360) * 2πr
Given the length of arc as 115π/6 in and angle of 138°, hence:
Length of arc = (Ф/360) * 2πr
Substituting:
115π/6 = (138/360) * 2πr
r = 25 inches
The radius of the circle is 25 inches.
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7. At age 20, Heather began investing $3000 annually
into an account earning 7.5% interest compounded
annually. Lesley invested $6000 annually into a similar
account but began at age 40. They both stopped
contributing at age 65.
a) How much money did Heather and Lesley contribute
to their account?
b) What is the value of each of their investments when
they are 65 years old?
c) At age 65, when the investments mature, who has
more money and by how
much?
a) Heather contributed $135,000 and Lesley contributed $150,000 to their accounts.
b) Heather's investment is approximately $273,714.17, while Lesley's investment is approximately $191,048.18 when they are 65 years old.
c) Heather has more money by approximately $82,665.99 at age 65.
a) To find out how much money Heather and Lesley contributed to their accounts, we need to calculate the total contributions made by each of them.
Heather:
Heather started investing at age 20 and stopped at age 65, contributing $3000 annually. The number of years she contributed is (65 - 20) = 45 years.
Total contributions by Heather = $3000 × 45 = $135,000.
Lesley:
Lesley started investing at age 40 and stopped at age 65, contributing $6000 annually. The number of years she contributed is (65 - 40) = 25 years.
Total contributions by Lesley = $6000 × 25 = $150,000.
Therefore, Heather contributed $135,000 and Lesley contributed $150,000 to their respective accounts.
b) To calculate the value of their investments at age 65, we can use the formula for compound interest:
Future Value = Principal × (1 + interest rate)^number of years
Heather:
Principal (initial investment) = $3000
Interest rate = 7.5% = 0.075 (converted to decimal)
Number of years = 65 - 20 = 45
Future Value of Heather's investment = $3000 × (1 + 0.075)^45
Lesley:
Principal (initial investment) = $6000
Interest rate = 7.5% = 0.075 (converted to decimal)
Number of years = 65 - 40 = 25
Future Value of Lesley's investment = $6000 × (1 + 0.075)^25
Calculating these values:
Future Value of Heather's investment = $3000 × (1.075)^45 ≈ $273,714.17
Future Value of Lesley's investment = $6000 × (1.075)^25 ≈ $191,048.18
c) To determine who has more money at age 65 and by how much, we compare the future values of their investments.
Heather's investment value at age 65 = $273,714.17
Lesley's investment value at age 65 = $191,048.18
Therefore, Heather has more money at age 65, and the difference in their investments is approximately $273,714.17 - $191,048.18 = $82,665.99.
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Select all the statements that are true for the following systems of equations.
System A
2x-3y = 4
4x - y = 18
00
System B
3x - 4y = 5
y = 5x +3
All three systems have different solutions.
Systems B and C have the same solution.
System C simplifies to 2x-3y=4 and 4x-y=18 by dividing the second equation by three.
Systems A and B have different solutions.
Systems A and C have the same solution.
Reset
System C
2x-3y=4
12x-3y = 54
Next
The statements that are true about the system of equations are: Options C, D, and E.
How to Find the Solution to a Systems of Equations?Let's analyze each statement and determine whether it is true or false for the given systems of equations:
System A
2x - 3y = 4
4x - y = 18
System B
3x - 4y = 5
y = 5x + 3
System C
2x - 3y = 4
12x - 3y = 54
A. All three systems have different solutions.
To determine if the systems have different solutions, we need to solve them. Solving system A gives the solution x = 5 and y = -6. Solving system B gives the solution x = -1 and y = -2. Solving system C gives the solution x = 5 and y = -6. Therefore, this statement is false because systems A and C have the same solution.
B. Systems B and C have the same solution.
As mentioned above, solving system B gives the solution x = -1 and y = -2. Solving system C gives the solution x = 5 and y = -6. Therefore, this statement is false because systems B and C have different solutions.
C. System C simplifies to 2x-3y=4 and 4x-y=18 by dividing the second equation by three.
To simplify system C, we can divide the second equation by 3, resulting in:
2x - 3y = 4
4x - y = 18
This is exactly the same as system A. Therefore, this statement is true.
D. Systems A and B have different solutions.
As mentioned earlier, solving system A gives the solution x = 5 and y = -6. Solving system B gives the solution x = -1 and y = -2. Therefore, this statement is true.
E. Systems A and C have the same solution.
As mentioned earlier, solving system A gives the solution x = 5 and y = -6. Solving system C gives the solution x = 5 and y = -6. Therefore, this statement is true.
In summary:
A. False
B. False
C. True
D. True
E. True
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Complete Question:
Select all the statements that are true for the following systems of equations.
System A
2x - 3y = 4
4x - y = 18
System B
3x - 4y = 5
y = 5x +3
System C
2x - 3y = 4
12x - 3y = 54
A. All three systems have different solutions.
B. Systems B and C have the same solution.
C. System C simplifies to 2x-3y=4 and 4x-y=18 by dividing the second equation by three.
D. Systems A and B have different solutions.
E. Systems A and C have the same solution.