Using the quadratic formula to find the solutions, we get x = (-10 ± √40)/6
Using the quadratic formula to find the solutions to the equation.From the question, we have the following parameters that can be used in our computation:
3x² + 10x + 5=0
Using the quadratic formula, we have
x = (-10 ± √(10² - 4 * 3 * 5))/(2 * 3)
Evaluate the products and the exponents
So, we have
x = (-10 ± √40)/6
Hence, the solution is x = (-10 ± √40)/6
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What is the common ratio?
n f(n)
1 300
2 375
3 468.75
4 585.9375
Write an explicit rule for the geometric sequence
What is f(12)?
The common ratio is 1.25. An explicit rule for the geometric sequence is f(n) = 300(1.25)ⁿ⁻¹ . The value of f(12) is 5,722.05.
To find the common ratio of the sequence, we need to divide each term by the previous term. For example, to find the common ratio between the first two terms:
375/300 = 1.25
Similarly, we can find the common ratio between the second and third terms:
468.75/375 = 1.25
And the common ratio between the third and fourth terms:
585.9375/468.75 = 1.25
Since the common ratio is the same for each pair of adjacent terms, we can conclude that the explicit rule for the geometric sequence is:
f(n) = 300(1.25)ⁿ⁻¹
To find f(12), we can simply substitute 12 for n in the formula:
f(12) = 300(1.25)¹²⁻¹
f(12) = 300(1.25)¹¹
f(12) = 300(19.0735)
f(12) = 5,722.05
Therefore, f(12) is 5,722.05.
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I INCLUDED THE GRAPH! PLEASE HELP ITS URGENT PLEASE I AM DOING MY BEST TO RAISE MY GRADE!!!
Graph g(x)=−|x+3|−2.
Use the ray tool and select two points to graph each ray.
The graph of the function g(x) = −|x + 3| − 2 is added as an attachment
How to determine the graph of the functionFrom the question, we have the following parameters that can be used in our computation:
g(x) = −|x + 3| − 2
The above expression is an absolute value function that hs the following properties
Reflected over the x-axisTranslated left by 3 unitsTranslated down by 2 unitsVertex = (-3, -2)Next, we plot the graph
See attachment for the graph of the function
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A window frame is made of four inner squares like shown below.
Pleaseee helpp
The perimeter of the outer square in red is: 320 cm
What is the perimeter of the square?The perimeter of a square is defined by the formuls:
P = 4 * side length
Now, we are told that each of the internal 4 squares have a perimeter of 160 cm.
Thus:
160 = 4 * side length
side length = 160/4
side length = 40 cm
Now, this means that the side length of the outer square in red is:
Side length = 2 * 40
= 80 cm
Thus:
Perimeter of outer square in red = 4 * 80
= 320 cm
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A company produces ceramic floor tiles which are supposed to have a surface area of 16 square inches. Due to variability in the manufacturing process, the actual surface area has a normal distribution with mean 16.1 square inches and standard deviation 0.2 square inches. What is the proportion of tiles produced by the process with surface area less than 16.0 square inches?
To find the proportion of tiles produced with a surface area less than 16.0 square inches, we'll use the properties of the normal distribution. Here are the steps:
1. Identify the given information: mean (μ) = 16.1 square inches, standard deviation (σ) = 0.2 square inches, and the desired surface area (x) = 16.0 square inches.
2. Calculate the z-score using the formula: z = (x - μ) / σ
z = (16.0 - 16.1) / 0.2
z = (-0.1) / 0.2
z = -0.5
3. Look up the z-score (-0.5) in a standard normal distribution table or use a calculator to find the area to the left of the z-score. This area represents the proportion of tiles with a surface area less than 16.0 square inches.
4. The area to the left of z = -0.5 is approximately 0.3085.
So, the proportion of tiles produced with a surface area less than 16.0 square inches is approximately 0.3085, or 30.85%.
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At a craft shop, a painter decided to paint a welcome sign to take home. An image of the sign is shown.
A five-sided figure with a flat top labeled 5 and one-half feet. A height labeled 4 feet. The length of the entire image is 9 ft. There is a point coming out of the right side of the image that is created by two line segments.
What is the area of the sign?
19 square feet
22 square feet
29 square feet
36 square feet
The area of the composite figure is 29 feet squared.
How to find the area of a composite figure?A five-sided figure with a flat top labelled 5 and one-half feet. A height labelled 4 feet. The length of the entire image is 9 ft.
Therefore, the area of the composite figure can be found as follows;
The figure can be divide into two shapes which are rectangle and a triangle.
Hence,
area of the composite figure = area of the rectangle + area of the triangle
area of the rectangle = 4 × 5.5 = 22 ft²
area of the triangle = 1 / 2 bh
where
b = base h = heightarea of the triangle = 1 / 2 × 4 × (9 - 5.5)
area of the triangle = 1 / 2 × 4 × 3.5
area of the triangle = 14 / 2
area of the triangle = 7 ft²
Therefore,
area of the composite figure = 22 + 7
area of the composite figure = 29 ft²
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You pay $1 to play a game in which you roll one fair die. If you roll a 6 on the first roll, you win $5. If you roll a 1 or a 2, you win $2. If not, you lose money.
a. Start with $10. Play the game 10 times. Keep track of the number of times you win and determine the amount of money you have left, at the end of the game.
b. Create a probability distribution for this game.
c. Find the expected value for this game.
After 10 rolls, we won 3 times and lost 7 times, and we have $11 left.
The probability distribution for this game is:
Outcome Probability
Lose 2/3
Win $2 1/6
Win $5 1/6
How to explain the probabilityIt should be noted that to calculate the anticipated value, multiply the likelihood of each scenario by its payment and add them together:
E(X) = (2/3) * (-1) + (1/6) * 2 + (1/6) * 5 = -2/3 + 1/3 + 5/6 = 1/2
As a result, the expected value of this game is $0.50. This indicates that if you play it frequently, you can expect to win $0.50 each game on average. However, you could win or lose money in any particular game.
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No current will flow between two charged bodies if they have the
same
A) resistance
B) charge
C) potential
D) charge/ potential ratio
Two bodies can have the same resistance or charge/potential ratio, but still have different potentials, resulting in the flow of current between them.
The correct answer is C) potential.
When two bodies have the same potential, it means that the electric potential difference between them is zero. In this case, no work is required to move a charge from one body to the other, because the potential energy of the charge is the same on both bodies.
Since current is defined as the flow of electric charge, if there is no potential difference between two bodies, there will be no force driving the charges to move from one body to the other. Hence, no current will flow between the two bodies.
It is important to note that having the same resistance or charge/ potential ratio does not necessarily mean that no current will flow between two bodies. Resistance refers to the opposition to the flow of current, and the charge/ potential ratio is the charge per unit of electric potential. Therefore, two bodies can have the same resistance or charge/potential ratio, but still have different potentials, resulting in the flow of current between them.
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PLEASE ANSWER!!!! 20 POINTS
--
Find the mean x of the data 16, 31, 38, 24, 36
Answer:
Find the mean x of the data 16, 31, 38, 24, 36
16 + 31 + 38 + 24 + 36
= 145
145 ÷ 5
= 29Step-by-step explanation:
You're welcome.
Find the slope for the line that passes through the points (-2,5) and (1,0)
Answer:
[tex]m=\frac{-5}{3}[/tex]
Step-by-step explanation:
Pre-SolvingWe want to find the slope between the points (-2,5) and (1,0).
The slope (m) can be found using the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] are points.
SolvingWe are already given the values of the points, but let's label their values to avoid any confusion and mistakes.
[tex]x_1=-2\\y_1=5\\x_2=1\\y_2=0[/tex]
Now, substitute into the formula.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{0-5}{1--2}[/tex]
Simplify this to:
[tex]m=\frac{0-5}{1+2}[/tex]
[tex]m=\frac{-5}{3}[/tex]
The slope is -5/3.
8. Two friends, Suban and Jas, start from the same point but ride their bikes down two different paths. The paths diverge at an angle of 38º. Suban rides down his path at 12 km/h, and Jas rides down his path at 14 km/h. Determine how far apart the two are after riding at their respective speeds for 90 min. (App/4) Problem: Show all steps of your work for full credit. 9. Gursant and Leo are both standing on the north side of a monument that is 6.0 m tall. Leo is standing 3.5 m closer to the monument than Gursant. Leo measures the angle from the ground to the top of the monument to be 41º. Determine the angle that Gursant would measure from the ground to the top of the monument, to the nearest degree. (App/4)
Gursant would measure an angle of approximately 29º (rounded to the nearest degree) from the ground to the top of the monument.
To solve this problem, we will use the formula d = rt, where d is the distance traveled, r is the rate (or speed), and t is the time.
First, we need to convert the time of 90 minutes to hours by dividing by 60: 90/60 = 1.5 hours.
Let's call the distance traveled by Suban "d1" and the distance traveled by Jas "d2". We want to find the distance between them, which we can call "d".
Using trigonometry, we can find that d1 = 12*cos(38º)1.5 = 11.513 km (rounded to 3 decimal places) and d2 = 14sin(38º)*1.5 = 8.307 km (rounded to 3 decimal places).
Now we can find the distance between them using the Pythagorean theorem: d^2 = d1^2 + d2^2.
Plugging in the values we found, we get d^2 = (11.513 km)^2 + (8.307 km)^2 = 221.93 km^2.
Taking the square root of both sides, we get d = 14.899 km (rounded to 3 decimal places).
Therefore, the two friends are approximately 14.899 km apart after riding for 90 minutes.
Let's call the distance from Gursant to the monument "x". Since Leo is standing 3.5 m closer to the monument, his distance from the monument is "x - 3.5".
Using trigonometry, we can set up the following equation:
tan(41º) = 6/x
Solving for x, we get:
x = 6/tan(41º) = 7.967 m (rounded to 3 decimal places)
Now we can find the angle that Gursant would measure from the ground to the top of the monument using the following equation:
tan(theta) = 6/(x + 3.5)
Plugging in the value we found for x, we get:
tan(theta) = 6/11.467
Solving for theta, we get:
theta = arctan(6/11.467) = 28.52º (rounded to the nearest degree)
Therefore, Gursant would measure an angle of approximately 29º (rounded to the nearest degree) from the ground to the top of the monument.
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The plane passing through the point P(1,3,4) with normal vector 2i+63 +7k has equation 5+3y+42=48 · Answer Ο Α True O B False
The statement "The plane passing through the point P(1,3,4) with normal vector 2i+63 +7k has equation 5+3y+42=48" is false because the equation '5+3y+42=48' given in the question is wrong.
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Now answer the question:
Claire and her children went into a grocery store and she bought $8 worth of apples
and bananas. Each apple costs $1 and each banana costs $0.50. She bought a total of
11 apples and bananas altogether. Determine the number of apples, x, and the
number of bananas, y, that Claire bought.
So if she bought a total of $8 worth that means there is more than one possibility but it says apples and bananas total but I’m gonna do more than that
For a total of $8 she could by 16 bananas and 0 apples
For $8 she could by 8 apples and zero bananas
For $8 she could by 4 apples and 8 bananas
Which expression is equivalent to x^{2}-36
The answer is
(-x-6i)(x-6i)
Show that the functions f(x) = x, g(x) = x - 1, and h(x) = x + 3 are linearly dependent. Show, however, that f(x) = x2, g(x) = x - 1, and h(x) = x + 3 are linearly independent
To show that the functions f(x) = x, g(x) = x - 1, and h(x) = x + 3 are linearly dependent, we need to find a non-zero linear combination of the three functions that equals zero.
Let's assume that a, b, and c are constants such that:
a*f(x) + b*g(x) + c*h(x) = 0
Substituting in the given functions, we get:
a*x + b*(x - 1) + c*(x + 3) = 0
Simplifying this equation, we get:
(a + b + c) * x + (-b + 3c) = 0
For this equation to hold true for all x, we must have:
a + b + c = 0
-b + 3c = 0
This is a system of two equations with three unknowns, which means that we have infinitely many solutions. For example, we could choose a = 1, b = -2, and c = 1, and the equation would hold true. Therefore, we have shown that the functions f(x) = x, g(x) = x - 1, and h(x) = x + 3 are linearly dependent.
Now, let's show that the functions f(x) = x^2, g(x) = x - 1, and h(x) = x + 3 are linearly independent.
We need to show that there are no non-zero constants a, b, and c such that:
a*f(x) + b*g(x) + c*h(x) = 0
Substituting in the given functions, we get:
a*x^2 + b*(x - 1) + c*(x + 3) = 0
This equation holds true for all x if and only if its coefficients are all zero. Therefore, we need to solve the system of three equations:
a = 0
-b + c = 0
3c = 0
The first equation tells us that a must be zero. The third equation tells us that c must be zero. Substituting c = 0 into the second equation, we get:
-b = 0
Therefore, we must have b = 0 as well.
Since a, b, and c are all zero, we have shown that the functions f(x) = x^2, g(x) = x - 1, and h(x) = x + 3 are linearly independent.
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38. A new apartment complex with 90 one-bedroom apartment units and 100 two-bedroom apartment units was built near a lake. Rental prices that will provide full occupancy are estimated at $1200 for one-bedroom units and $1800 for two-bedroom units. A market survey suggests that for every $20 increase in the price of a one-bedroom unit one less customer will sign a lease and for every $60 increase in the price of a two-bedroom unit two less customers will sign a lease. What rental price should the manager charge to maximize revenue?
The required manager should charge $1600 for one-bedroom units and $2250 for two-bedroom units to maximize revenue.
Let x be the number of $20 increases in the price of a one-bedroom unit, and y be the number of $60 increases in the price of a two-bedroom unit. Then the rental prices for one-bedroom and two-bedroom units can be expressed as:
One-bedroom price = $1200 + $20x
Two-bedroom price = $1800 + $60y
The total number of customers for one-bedroom units is 90 minus the number of customers lost due to the price increase, which is x. Similarly, the total number of customers for two-bedroom units is 100 minus the number of customers lost due to the price increase, which is 2y. Therefore, the total revenue can be expressed as:
Revenue = (90 - x) * ($1200 + $20x) + (100 - 2y) * ($1800 + $60y)
Expanding and simplifying this expression, we get:
Revenue = 216000 + 9600x - 240x² + 180000 + 108000y - 7200y²
Collecting like terms, we get:
Revenue = -240x² - 7200y² + 9600x + 108000y + 396000
To find the rental price that maximizes revenue, we need to find the values of x and y that maximize the revenue. We can do this by taking partial derivatives of the revenue function with respect to x and y and setting them equal to zero:
dRevenue/dx = -480x + 9600 = 0
dRevenue/dy = -14400y + 108000 = 0
Solving for x and y, we get:
x = 20
y = 7.5
Therefore, the rental prices that maximize revenue are:
One-bedroom price = $1200 + $20x = $1600
Two-bedroom price = $1800 + $60y = $2250
So the manager should charge $1600 for one-bedroom units and $2250 for two-bedroom units to maximize revenue.
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The tree diagram represents an
experiment consisting of two trials.
S
A
B
.4 C
6
13
D
C
D
The required probability is P(A and C) is 0.2 which is represented in the tree diagram.
What is probability?Probability is defined as the possibility of an event being equal to the ratio of the number of favorable outcomes and the total number of outcomes.
The given tree diagram represents an experiment consisting of two trials.
The tree diagram represents an experiment consisting of two trials. In this case, the probability of event A and event C occurring is represented by the intersection of branches A and C in the tree diagram.
This probability can be calculated by multiplying the probability of each individual event together.
As per the given question, we have
P(A) = 0.5
P(C|A) = 0.4
So, P(A and C) = 0.5 × 0.4 = 0.2
Thus, the required probability is P(A and C) is 0.2
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Michael has set up an IRA and will deposit $3,000 at the end of each year from age 25 to age 65. Find the
amount of the annuity if the investment is in a stock fund yielding 7% interest, compounded annually.
$300,000.00
$199,635.28
$598,905.30
$226, 351.17
The amount (future value) of the annuity, if a $3,000 annual deposit is made in a stock fund yielding 7% interest, compounded annually, is C. $598,905.30.
How the future value is determined:The future value is determined by compounding the periodic deposits and interests.
Compounding describes a process that charges interest on interest.
The future value can be computed using an online finance calculator as follows:
N (# of periods) = 40 years (65 - 25)
I/Y (Interest per year) = 7%
PV (Present Value) = $0
PMT (Periodic Payment) = $3,000
Results:
Future Value (FV) = $598,905.34
The sum of all periodic payments = $120,000.00
Total Interest = $478,905.34
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Answer Immediaetly Please
Given SV = 15, UV = 30, and RS = 55, we found TV by using the fact that triangles TRU and SUC are similar. The length of TV is 110.
In the given diagram, we have a triangle TRS with a line UV that is parallel to the base RS. We are given that SV = 15, UV = 30, and RS = 55, and we need to find the length of TV.
To find TV, we can use the fact that UV is parallel to RS, which means that triangles TRU and SUV are similar.
Using the similarity of triangles TRU and SUC, we can set up the following proportion
TV / RS = UV / SV
Substituting the given values
TV / 55 = 30 / 15
Simplifying
TV / 55 = 2
Multiplying both sides by 55
TV = 110
Therefore, the length of TV is 110.
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a recent study at a university showed that the proportion of students who commute more than 15 miles to school is 25%. suppose we have good reason to suspect that the proportion is greater than 25%, and we carry out a hypothesis test. state the null hypothesis h0 and the alternative hypothesis h1 that we would use for this test.H0:H1:
Answer:
las cañaverales son extenso y hay numerosos
The null hypothesis, H0, is that the proportion of students who commute more than 15 miles to school is equal to or less than 25%. The alternative hypothesis, H1, is that the proportion is greater than 25%.
H0: Proportion of students who commute more than 15 miles to school ≤ 25%
H1: Proportion of students who commute more than 15 miles to school > 25%
In this hypothesis test, we will be using the following terms:
- Null Hypothesis (H0): The proportion of students who commute more than 15 miles to school is equal to 25%.
- Alternative Hypothesis (H1): The proportion of students who commute more than 15 miles to school is greater than 25%.
To restate the hypotheses:
H0: p = 0.25
H1: p > 0.25
Here, p represents the proportion of students who commute more than 15 miles to school.
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spinner has 4 red 3 blues and one yello what is the theoretical probability that it will spin red then blue
The theoretical probability of the spinner landing on red then blue is 1/9
Determining the theoretical probability of the spinner landing on red then blueFrom the question, we have the following parameters that can be used in our computation:
Colors = 3 i.e. yellow, blue and red
Blue = 1
Red = 1
So, we have
Theoretical probability = Red/Colors * Blue/Colors
Substitute the known values in the above equation, so, we have the following representation
Theoretical probability = 1/3 * 1/3
Evaluate
Theoretical probability = 1/9
Hence, theoretical probability is 1/9
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Brewsky's is a chain of micro-breweries. Managers are interested in the costs of the stores and believe that the costs can be explained in large part by the number of customers patron¬izing the stores. Monthly data regarding customer visits and costs for the preceding year for one of the stores have been entered into the regression analysis and the analysis is as follows:Average monthly customer-visits 1,462Average monthly total costs $ 4,629Regression Results Intercept $ 1,496b coefficient $ 2.08R2 0.868141. In a regression equation expressed as y = a + bx, how is the letter b best described? (CMA adapted)a. The proximity of the data points to the regression line.b. The estimate of the cost for an additional customer visit.c.The fixed costs per customer-visit.d.An estimate of the probability of return customers.2. How is the letter x in the regression equation best described? (CMA adapted)a. The observed customer visits for a given month.b. Fixed costs per each customer-visit.c. The observed store costs for a given month.d. The estimate of the number of new customer visits for the month3. What is the percent of the total variance that can be explained by the regression equation? (CMA adapted)a. 86.8%b. 71.9%c. 31.6%d. 97.7%
In this regression analysis, the letter b in the equation y = a + bx represents the estimate of the cost for an additional customer visit. This means that for every additional customer visit to the store, the expected increase in monthly total costs is $2.08, according to the regression model.
The letter x in the regression equation represents the observed customer visits for a given month. This means that the regression model is predicting the monthly total costs based on the number of customer visits in that month.
The R2 value of 0.8681 means that 86.81% of the total variance in the monthly total costs can be explained by the regression equation, which indicates a strong relationship between the number of customer visits and the total costs. This can help managers of Brewsky's make informed decisions about how to allocate resources and improve profitability. However, it is important to note that other factors may also influence the costs, and the regression model may not capture all of these factors.
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Use the formula V = lwh to find the volume of the box.
23. I is 10 in., w is 8 in., and h is 5 in.
Answer:
400 in²
Step-by-step explanation:
We Know
The formula V = l · w · h
l = 10 in
w = 8 in
h = 5 in
Find the volume of the box.
We Take
10 · 8 · 5 = 400 in²
So, the volume of the box is 400 in²
volume of the box is 400 cubic inches.
The given values are:
l = 10 inches
w = 8 inches
h = 5 inches
Using the formula for the volume of a box:
V = lwh
Substituting the given values:
V = (10 in.)(8 in.)(5 in.)
Multiplying:
V = 400 cubic inches
Therefore, the volume of the box is 400 cubic inches.
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a bowl contains three red and four yellow marbles. you randomly select two marbles from the bowl. which of the following is a conditional probability? assume the second marble is drawn from the marbles remaining after the first draw.
The conditional probability in this scenario is the probability of drawing a yellow marble on the second draw, given that the first marble drawn was red.
To calculate this conditional probability, we can use Bayes' theorem, which states that the probability of an event (in this case, drawing a yellow marble on the second draw) given some prior knowledge (in this case, that the first marble drawn was red) is equal to the probability of both events occurring (drawing a red marble first and a yellow marble second) divided by the probability of the prior event (drawing a red marble first).
The probability of drawing a red marble first is 3/7 since there are three red marbles out of a total of seven marbles in the bowl. Once a red marble is drawn, there are six marbles remaining, of which three are yellow. Therefore, the probability of drawing a yellow marble second, given that the first marble was red, is 3/6 or 1/2.
Putting this together, we can calculate the conditional probability as follows:
P(Yellow on Second Draw | Red on First Draw) = P(Red on First Draw and Yellow on Second Draw) / P(Red on First Draw)
= (3/7) * (3/6) / (3/7)
= 1/2
Therefore, the conditional probability in this scenario is 1/2 or 50%. This means that there is a 50% chance of drawing a yellow marble on the second draw, given that the first marble drawn was red.
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A paper bag has seven colored marbles. The marbles are pink, red, green, blue, purple, yellow, and orange. List the sample space when choosing one marble.
S = {1, 2, 3, 4, 5, 6}
S = {purple, pink, red, blue, green, orange, yellow}
S = {g, r, b, y, o, p}
S = {green, blue, yellow, orange, purple, red}
the answer to your math question is S = {green, blue, yellow, orange, purple, red}
The square of a positive number is 42 more than the number itself. What is the number?
The number we're looking for is 7.
Let's call the number we're looking for "x". According to the problem, the square of the number is 42 more than the number itself. In equation form, this can be written as:
[tex]x^2[/tex] = x + 42
To solve for x, we want to get all the terms on one side of the equation. We can start by subtracting x + 42 from both sides:
[tex]x^2[/tex] - x - 42 = 0
Now we have a quadratic equation. We can solve it by factoring or by using the quadratic formula. Let's use factoring. We want to find two numbers that multiply to -42 and add up to -1 (since the coefficient of x is -1). One possible pair of numbers is -7 and 6, since -7 × 6 = -42 and -7 + 6 = -1. So we can rewrite the equation as:
(x - 7)(x + 6) = 0
This tells us that either x - 7 = 0 or x + 6 = 0. Solving for x in each case, we get:x = 7 or x = -6
We're looking for a positive number, so the solution is x = 7. Therefore, the number we're looking for is 7.
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A store sells ink cartridges in packages.
Ink World Packages
Number of Cartridges
Total Cost
Package A
3
$60
Package B
6
$60
Package C
1
$20
Package D
3
$20
Which two packages have the same ratio of cartridges to cost?
For a store sells ink cartridges in packages, two packages have the same ratio of cartridges to cost are Package A and Package C.
A ratio is used to comparison of two quantities. An equivalent or same ratio means a ratio that is equal to or has the same value as another ratio. We have a store sells ink cartridges in packages. The table represents the ink cartridges, number Cartridges and total cost.
Ink World Number of Total Cost
Packages Cartridges
Package A 3 $60
Package B 6 $60
Package C 1 $20
Package D 3 $20
We have to determine two packages have the same ratio of cartridges to cost.
Now, check the ratio of cartridges to cost for each packages. For package A,
cartridges : cost = 3 : 60 = 1 : 20
For package B, cartridges : cost = 6 : 60 = 1 : 10
For package C, cartridges : cost = 1 : 20
For package D, cartridges : cost = 3 : 20 = 3 : 20
So, the packagses with same ratio are package A and C.
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unit systems of equations homework 5
Solving a system of equations necessarily necessitates the utilization of an appropriate unit system, depending on the equations to be solved.
How to explain the equationHere are a few common unit systems employed when attempting to resolve such equations:
Metric System: This involves adhering to the International System of Units (SI), which is used across the world; in this premise, meters, kilograms, and seconds depict length, mass, and time respectively.
Imperial System: Proffered mainly in the United States, this method applies units like feet, pounds, and seconds for sizing, mass and time correspondingly.
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suppose there is a lottery where the organizers pick a set of 11 distinct numbers. a player then picks 7 distinct numbers and wins when all 7 are in the set chosen by the organizers. numbers chosen by both the players and organizers come from the set {1, 2, ..., 80}. (a) let the sample space, s, be all the sets of 7 numbers the player can choose. what is |s|? (b) let e be the event that all the numbers the player chooses are in the winning set. what is |e|? (c) what is the probability of winning? as a reminder, you may leave your answer un- simplified.
(a) 40,475,358.
(b) 330
(c) 0.0008%.
(a) To find |S|, the total number of sets of 7 distinct numbers a player can choose, we need to find the combinations of choosing 7 numbers from the 80 available options. This can be calculated using the combination formula:
C(n, k) = n! / (k! * (n - k)!)
In this case, n = 80 (total numbers) and k = 7 (numbers to choose). So, |S| = C(80, 7):
|S| = 80! / (7! * (80 - 7)!)
|S| = 80! / (7! * 73!)
(b) To find |E|, the number of sets where all 7 numbers chosen by the player are in the winning set of 11 numbers chosen by the organizers, we need to find the combinations of choosing 7 numbers from the 11 available options in the winning set:
|E| = C(11, 7)
|E| = 11! / (7! * (11 - 7)!)
|E| = 11! / (7! * 4!)
(c) To find the probability of winning, we need to calculate the ratio of the favorable outcomes (|E|) to the total possible outcomes (|S|):
P(winning) = |E| / |S|
P(winning) = (11! / (7! * 4!)) / (80! / (7! * 73!))
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Charles inherited $500,000 from his grandfather. He decides to invest the money into a fund that has a
4% annual interest.
9. If the interest is compounded continuously, what is the total amount in his account after ten years?
10. How much more does Charles earn after ten years by putting his investment into an account that is
compounded continuously versus an account that is compounded annually?
11. Approximately how long will it take for his account balance to double?
a) The total amount in Charles' account after ten years is $728,215.72.
b) Charles earns $29,375.31 more by putting his investment into an account that is compounded continuously instead of annually.
c) It will take approximately 17.3 years for Charles' account balance to double.
a. To calculate the total amount in Charles' account after ten years, we use the formula for continuous compound interest:
A = [tex]Pe^{(rt)[/tex]
where A is the amount in the account after t years, P is the initial principal, r is the annual interest rate as a decimal, and e is the constant approximately equal to 2.71828.
Substituting the given values, we get:
A = 500,000[tex]e^{(0.0410)[/tex]
A = $728,215.72
b. To calculate the difference in earnings between continuous and annual compounding, we use the formula:
A = P(1 + r/n)^(nt)
where n is the number of times interest is compounded per year. For continuous compounding, n approaches infinity, so the formula becomes:
A = [tex]Pe^{(rt)[/tex]
Substituting the given values, we get:
A = 500,000[tex]e^{(0.0410)[/tex]
A = $728,215.72
For annual compounding, n = 1, so the formula becomes:
Al = 500,000*[tex](1 + 0.04/1)^{(1*10)[/tex]
Al = $698,840.41
Therefore, the difference in earnings between continuous and annual compounding is:
A - Al = $728,215.72 - $698,840.41 = $29,375.31
c. To find the time it takes for Charles' account balance to double, we use the formula:
A = [tex]Pe^{(rt)[/tex]
We want to find t when A = 2P, so we can write:
2P = [tex]Pe^{(0.04t)[/tex]
Dividing both sides by P and taking the natural logarithm, we get:
ln(2) = 0.04t
Solving for t, we get:
t = ln(2)/0.04
t ≈ 17.3 years
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A 12-foot pole is supporting a tent and has a rope attached to the top
The expression that represent the length of the rope is 10 / cos 40° = 13.1 feet
How to find the expression that show the length of the rope?A 12-foot pole is supporting a tent and has a rope attached to the top. The rope is pulled straight and the other end is attached to a peg two foot above the ground.
This situation forms a right angle triangle. Therefore, let's find the expression that shows the length of the rope using trigonometric ratios.
Hence,
cos 40 = adjacent / hypotenuse
adjacent side = 10 ft
Therefore,
cos 40° = 10 / x
where
x = length of the ropecross multiply
x = 10 / cos 40°
x = 10 / 0.76604444311
x = 13.0548302872
x = 13.1 feet
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