Answers
Completing the square
(x+5)^2 - 74
Related Questions
The hypotenuse of a right triangle measures 29 cm. One leg is 1 cm shorter than the other. What are the lengths of the legs?
Answers
The length of the legs are 20 cm and 21 cm
What is the length of the legs?The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
We know that from the Pythagoras theorem;
[tex]c^2 = a^2 + b^2[/tex]
Let the hypotenuse be c and the other two sides be a and b
We have that;
[tex]29^2 = x^2 + (x -1)^2\\841 = x^2 + x^2 - 2x + 1\\841 = 2x^2 - 2x + 1\\2x^2 - 2x + 1 - 841 = 0\\2x^2 - 2x - 840 = 0\\x = -20 or 21[/tex]
Since length can not be negative, x = 21 cm
Thus the other leg is 20 cm
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What is the 5280th digit in the decimal expansion of 5/17
Answers
The second digit after the decimal point is 9. We repeat this process until we have found the 5280th digit:
```
0.294117647058823529...
50
Calculate the decimal expansion?To find the 5280th digit in the decimal expansion of 5/17, we need to find the first 5280 digits of the decimal expansion and then look at the 5280th digit.
To do this, we can use long division to divide 5 by 17. We start by dividing 5 by 17 to get the first digit after the decimal point:
```
0.294117647058823529...
```
We can see that the first digit after the decimal point is 2. To get the second digit, we multiply the remainder (5) by 10 and then divide by 17:
```
5 * 10 = 50
50 / 17 = 2 remainder 16
```
The second digit after the decimal point is 9. We repeat this process until we have found the 5280th digit:
```
0.294117647058823529...
50
-----
5 * 10 = 50 | 16.0000000000000000000000000000000000000000000000000000000000000000000000...
0
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For evrey 500g of reactants, 3. 1 g of catalyst were required. How much catalyst was required for 900g of reactants
Answers
5.58g of catalyst is required for 900g of reactants.
How much catalyst for 900g reactants?If 500g of reactants require 3.1g of catalyst, then for 900g of reactants, we can use the following proportion:
500g reactants / 3.1g catalyst = 900g reactants / x
Where x is the amount of catalyst required for 900g of reactants.
To solve for x, we can cross-multiply:
500g reactants * x = 3.1g catalyst * 900g reactants
Then, we can divide both sides by 500g reactants to isolate x:
x = (3.1g catalyst * 900g reactants) / 500g reactants
Simplifying this expression gives:
x = 5.58g catalyst
Therefore, 5.58g of catalyst is required for 900g of reactants.
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Use the given acceleration function and initial conditions to find the velocity vector v(t), and position vector r(t). Then find the position at time t = 9. a(t) = −cos ti − sin tj v(0) = j + k, r(0) = i v(t) = r(t) = r(9) =
Answers
find the position at time t = 9. a(t) = −cos ti − sin tj v(0) = j + k, r(0) = i v(t) = r(t) = r(9) = This gives you the position vector r(9) as a function of sin(9) and cos(9).
To find the velocity vector v(t) and position vector r(t), we need to integrate the given acceleration function a(t) and apply the initial conditions. Here's a step-by-step explanation:
1. Given acceleration function: a(t) = -cos(t)i - sin(t)j
2. Integrate a(t) with respect to t to find v(t):
v(t) = ∫(-cos(t)i - sin(t)j) dt = (sin(t)i + cos(t)j) + C, where C is a constant vector.
3. Apply initial condition v(0) = j + k:
v(0) = sin(0)i + cos(0)j + C = j + k
C = -i + j + k
4. The velocity function is: v(t) = sin(t)i + cos(t)j - i + j + k
Now let's find the position vector r(t):
5. Integrate v(t) with respect to t to find r(t):
r(t) = ∫(sin(t)i + cos(t)j - i + j + k) dt = (-cos(t)i + sin(t)j + t(k) + D, where D is another constant vector.
6. Apply initial condition r(0) = i:
r(0) = -cos(0)i + sin(0)j + 0(k) + D = i
D = i
7. The position function is: r(t) = -cos(t)i + sin(t)j + tk + i
Finally, let's find the position at time t = 9:
8. r(9) = -cos(9)i + sin(9)j + 9k + i
This gives you the position vector r(9) as a function of sin(9) and cos(9).
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how to find the polynmial closest to another polynomial in an inner product space
Answers
To find the polynomial closest to another polynomial in an inner product space, you can follow these steps:
Choose an inner product on the space of polynomials. One common inner product on this space is the L2 inner product, which is defined as:
<f,g> = ∫a^b f(x)g(x) dx,
where a and b are the endpoints of the interval on which the polynomials are defined.
Let P be the space of polynomials of degree at most n, where n is the degree of the polynomial you want to approximate. Let f be the polynomial you want to approximate, and let g be an arbitrary polynomial in P.
Define the error between f and g as e = f - g.
Compute the inner product of e with itself:
<e,e> = ∫[tex]a^b (f(x) - g(x))^2 dx.[/tex]
Minimize this inner product with respect to g. This can be done by setting the derivative of <e,e> with respect to g equal to zero and solving for g.
The polynomial that minimizes the error is the polynomial closest to f in the L2 sense.
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Two friends larbi and aminu,plays a game of chess with equal amount of money at the beginning (zero sum games) at the end of the game larbi lost 5 elevens of his amount and aminu gains 6 cedis more than one half of what is left for larbi. what total amount of money was left at the beginning of the game
Answers
At the beginning of the game, the total amount of money between Larbi and Aminu was 66 cedis.
Let x be the total amount of money at the beginning of the game.
After the game, Larbi lost 5/11x, so he has (1-5/11)x = 6/11x left.
Aminu gained 6 more than 1/2 of what Larbi has left, which is (1/2)(6/11x) + 6 = 3/11x + 6.
The total amount left after the game is the sum of what Larbi and Aminu have, which is (6/11x) + (3/11x + 6) = (9/11x) + 6.
Since this is equal to x (the total amount they started with), we have:
(9/11x) + 6 = x
Solving for x, we get:
x = 66.
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Graph: 3y = 6
-
-6 -4 -2
6
4
2
-2
त्र
-6
y
2 4
st
Click or tap the graph to plot a point.
6
X
४
Draw
y
Answers
A circular mirror has a radius of 3. 4 feet rosalinda is decorating the edge of the mirror with Washington tape if she has exactly enough washi tape which measurement is closest to the length of the piece of washi tape in feet
Answers
The measurement closest to the length of the piece of washi tape needed is approximately 21.36 feet.
The circumference of the circular mirror can be calculated using the formula C = 2πr, where r is the radius. Plugging in the given radius of 3.4 feet, we get C = 2π(3.4) = 21.36 feet (rounded to two decimal places). Since Rosalinda is decorating the edge of the mirror with washi tape, she needs a piece of tape that is equal in length to the circumference of the mirror. Therefore, the length of the piece of washi tape needed is closest to 21.36 feet.
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You deposit $2000 earned at a summer job in an account that pays 4. 2% simple interest. What is the balance in the account in 3 years? Estimate to the nearest whole number
Answers
A deposit of $2000 earning 4.2% simple interest for 3 years will have a balance of $2252. The estimated balance rounded to the nearest whole number is $2252.
To calculate the balance in the account after 3 years, we can use the formula
balance = principal x (1 + interest rate x time)
Plugging in the values, we get
balance = 2000 x (1 + 0.042 x 3)
balance = 2000 x (1 + 0.126)
balance = 2000 x 1.126
balance = 2252
Therefore, the balance in the account after 3 years is $2252.
As for the estimate, since the interest is simple, we can approximate it by multiplying the interest rate by the number of years and adding it to the principal. So, the estimate would be
estimate = principal x (1 + interest rate x time)
estimate = 2000 x (1 + 0.042 x 3)
estimate = 2000 x (1 + 0.126)
estimate = 2000 x 1.126
estimate = 2252
Rounding to the nearest whole number, the estimate is also $2252.
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Please help it would be amazing if you knew this
Answers
The solution of the composite function, (f + g)(x) is 8x + 7
How to solve function?A function relates input and output. In other words, a function is a special relationship among the inputs (independent variable) and their outputs (dependent variable).
A composite function is a function that depends on another function.
Therefore, let's solve the composite function
f(x) = x + 3
g(x) = 7x + 4
Hence,
(f + g)(x) can be solved as follows:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = x + 3 + 7x + 4
combine like terms
(f + g)(x) = x + 7x + 3 + 4
(f + g)(x) = 8x + 7
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Answer the following questions using what you've learned from this lesson. Write your responses in the
space provided.
For questions 1-4, use the following data to calculate and interpret the linear regression equation
Worldwide carbon dioxide emissions have increased over the years as Earth's population has grown. The
table shows the world carbon dioxide emissions, in millions of metric tons, from 1950 to 1990
Year
1950
1960
Response variable:
1970
1980
1990
1. Which is the explanatory variable.
and which is the response
variable?
Explanatory variable:
Name
Date
Emissions
6000
9500
15,000
19,300
22,500
2. Calculate the linear regression
equation from the above data.
Answers
The expected value of emissions when Year is 0, which makes up the intercept of the line (-666708.3) .
what is linear regression ?By fitting a linear equation to the observed data, the statistical technique of linear regression is utilized to describe the connection between two variables. Finding the line that best fits the data and can accurately depict the interaction of the variables is the aim of linear regression. In regression analysis, one variable is regarded as the independent (or predictor) variable, and the other is regarded as the dependent (or response) factor (or response variable). Y = mx + b, where x is the underlying factor, m is the line's slope, and b is the y-intercept, is a linear equation that describes the connection between both variables.
given
The linear regression equation that results is:
Emissions = 666708.3 - 347.6 Year
This equation models the link between Year (the explanatory variable) and Emissions as a straight line (the response variable).
According to the line's slope (347.6), emissions rose by an average of 348 million metric tonnes annually during this time.
The expected value of emissions when Year is 0, which makes up the intercept of the line (-666708.3) .
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2. How many checks must a customer write per month before the new plan is cheaper than the old plan? and new plan? 3. What formula/equations can be formed to find the cost for any number of checks for the old cheaper for a customer who writes 10 checks per month? 1. Compute the cost of 10 checks under the old plan and under the new plan. Which plan is check will cost 8 cents. The bank claims the new plan will save the customer money. Plus 15 cents for each check announces that it will change its monthly fee to $3 and that each Problem #3 A bank that has been charging a monthly service fee of $2 for checking accounts
Answers
The old plan is cheaper for a customer who writes 10 checks per month.
To determine how many checks a customer must write per month before the new plan is cheaper than the old plan, we need to set up an equation to compare the two plans. Let x be the number of checks written per month. The cost of the old plan is given by:
C_old = 0.08x + 2
The cost of the new plan is given by:
C_new = 3 + 0.15x
To find out when the new plan becomes cheaper, we need to set the two costs equal to each other and solve for x:
0.08x + 2 = 3 + 0.15x
0.07x = 1
x ≈ 14.29
Therefore, a customer would need to write 15 checks per month for the new plan to be cheaper than the old plan.
For a customer who writes 10 checks per month, the cost of the old plan is:
C_old = 0.08(10) + 2 = 2.80
The cost of the new plan is:
C_new = 3 + 0.15(10) = 4.50
Therefore, the old plan is cheaper for a customer who writes 10 checks per month.
The formula for the cost of any number of checks for the old plan is:
C_old = 0.08x + 2
where x is the number of checks written per month
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A triangular prism is 15 feet long. It has a triangular face with a base of 10 feet the volume of the prism is 945 ft. What is the height of its triangular height
Answers
The height of the triangular face of a triangular prism with a length of 15 feet and a base of 10 feet, and a volume of 945 cubic feet is 12.6 feet."
The formula for the volume of a triangular prism is given by:
Volume = (1/2) x base x height x length
where base and height refer to the base and height of the triangular face, and length refers to the length of the prism.
Substituting the given values, we have:
945 = (1/2) x 10 x height x 15
Simplifying:
945 = 75 x height
Dividing both sides by 75:
height = 945/75
height = 12.6 feet
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A square has sides of length s. A rectangle is 6 inches shorter than the square and 1 inch longer. Which of the following expressions represents the perimeter of the rectangle?
Answers
The perimeter of the rectangle is represented by the expression 4s - 10.
How to calculate perimeter of a rectangle?
To calculate the perimeter of a rectangle, you need to add up the lengths of all four sides.
In the problem given, we know that the rectangle is 6 inches shorter than the square and 1 inch longer.
Let's call the length of the rectangle l and the width w.
We know that the length of the square is equal to its width (since it's a square), so the length of the rectangle must be l = s - 6, and the width must be w = s + 1.
To find the perimeter, we add up all four sides: P = 2l + 2w = 2(s-6) + 2(s+1) = 4s - 10.
Therefore, the expression that represents the perimeter of the rectangle is 4s - 10.
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HELPPPPPPPPPPP HELPPPP PLEASEEE ITS MATHHHH
Answers
sorry, I am really sorry I really need the points. Just by seeing it I think in the 3 one am not sure but don't trust me because I just saw it and thought it was 3........
7. At Burger Heaven a double contains 2 meat patties and 6 pickles, whereas a
triple contains 3 meat patties and 3 pickles. Near closing time one day, only
24 meat patties and 48 pickles are available. If a double burger sells for
$1. 20 and a triple burger sells for $1. 50, then how many of each should be
made to maximize the total revenue?
(4. 6 5pts)
a) Write your constraints (1pt)
Answers
At Burger Heaven, to maximize the total revenue from selling double burgers containing 2 meat patties and 6 pickles, you need to consider the following constraints:
1. Ingredient availability: Ensure that there are enough meat patties and pickles in stock to meet the demand for double burgers.
2. Production capacity: The kitchen staff must be able to efficiently prepare and assemble the double burgers without compromising on quality.
3. Pricing strategy: Set a competitive price for the double burger to attract customers and generate optimal revenue.
4. Demand forecasting: Accurately predict customer demand for the double burger to prevent overstocking or understocking of ingredients, which can impact revenue.
To maximize total revenue at Burger Heaven, follow these steps:
a) Analyze the availability of meat patties and pickles to determine how many double burgers can be made with the current inventory.
b) Evaluate the production capacity of the kitchen staff to ensure that they can efficiently prepare and assemble the double burgers.
c) Research the market to set a competitive price for the double burger, considering the costs of ingredients, labor, and other expenses.
d) Forecast customer demand for the double burger to ensure optimal inventory levels and to meet customer expectations.
By addressing these constraints and following the steps above, Burger Heaven can successfully maximize its total revenue from selling double burgers with 2 meat patties and 6 pickles.
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Find the equation for the line that:
passes through (-4,-7) and has slope -6/7
The slope intercept form of the function is:
Answers
Answer: [tex]y=\frac{-6}{7} x+\frac{-73}{7}[/tex]
Step-by-step explanation:
The slope intercept form for a line is y=mx+b, where m is slope and b is the y intercept. For this form, we need to know the slope and y intercept.
The slope and one x and y are give, so we can plug in all of these values into the slope intercept equation to solve for b.
Doing so, we get:
[tex]y=mx+b\\-7=\frac{-6}{7} (-4)+b\\b=-7-\frac{24}{7} \\b=\frac{-73}{7}[/tex]
So, knowing the slope and y intercept, our equation is
[tex]y=\frac{-6}{7} x+\frac{-73}{7}[/tex]
PLEASEEEE HELPPP ASAP 20 PTS
Use long division to determine the quotient of the following expression.
Write the quotient in standard form with the term of largest degree on the left. (10x^(2)+3x-77)-:(2x+7)
Answers
The quotient of the division 10x² + 3x - 77 ÷ 2x + 7 is 5x - 16
Evaluating the long division expressionsThe quotient expression is given as
10x² + 3x - 77 ÷ 2x + 7
The long division expression is represented as
2x + 7 | 10x² + 3x - 77
So, we have the following division process
5x - 16
2x + 7 | 10x² + 3x - 77
10x² + 35x
--------------------------------
-32x - 77
-32x - 112
-------------------------------------
35
Hence, the quotient of the long division is 5x - 16
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On the same coordinate plane, mark all points (x,y) such that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2
Answers
The points that satisfy equations (A), (B), and (C) are (-2,-4), (4,2), and (-4,2).
we can plot the graphs of each of these equations on the same coordinate plane and then identify the points where they intersect.
To mark all the points that satisfy the equations (A) [tex]y=x-2[/tex], (B) y=x-2[tex]y=x-2[/tex] and (C) [tex]y=|x|-2[/tex],
For equation (A), we can see that the slope is 1 (the coefficient of x) and the y-intercept is -2 (the constant term). This means that the graph of equation (A) is a straight line that passes through the point (0,-2) and has a slope of 1.
We can plot this line on the coordinate plane by marking the point (0,-2) and then drawing a line with slope 1 that passes through this point.
For equation (B), we can see that the slope is -1 (the coefficient of x) and the y-intercept is -2 (the constant term).
This means that the graph of equation (B) is a straight line that passes through the point (0,-2) and has a slope of -1. We can plot this line on the coordinate plane by marking the point (0,-2) and then drawing a line with slope -1 that passes through this point.
For equation (C), we can see that the y-intercept is -2 and that the graph of the equation is symmetric with respect to the y-axis.
This means that we only need to plot the part of the graph that lies in the first quadrant, and then we can use symmetry to find the part that lies in the other quadrants.
To plot the graph of equation (C) in the first quadrant, we can start by marking the point (2,0) (since y=|x|-2 when x=2) and then draw a V-shape with the vertex at this point and the arms of the V going up and to the right.
To find the points where these three graphs intersect, we can look for the points where any two of the graphs intersect. For example, we can see that the graphs of equations (A) and (B) intersect at the point (-2,-4).
Similarly, we can see that the graphs of equations (A) and (C) intersect at the point (4,2), and the graphs of equations (B) and (C) intersect at the point (-4,2).
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The polynomial
y
=
−
0.74
x
4
+
2.8
x
3
+
26.4
describes the billions of flu virus particles in a person’s body
x
days after being infected. find the number of virus particles, in billions, after 1 day.
Answers
The number of flu virus particles in a person's body after 1 day is 28.46 billion, based on the given polynomial function.
The polynomial given to us is, y = −0.74x⁴ + 2.8x³ + 26.4 which describes the billions of flu virus particles in a person’s body. To find the number of virus particles in a person's body after 1 day, we need to substitute x = 1 into the given polynomial and evaluate it.
So, we have,
y = -0.74(1)⁴ + 2.8(1)³ + 26.4
= -0.74 + 2.8 + 26.4
= 28.46
Therefore, the number of virus particles in a person's body after 1 day is 28.46 billion.
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Complete question - The polynomial y=−0.74x⁴ + 2.8x³ + 26.4 describes the billions of flu virus particles in a person’s body x days after being infected. find the number of virus particles, in billions, after 1 day.
A non-government that supports palay production in the philippines conducted the research that answer the question: is the proportion of palay harvested different from 0. 50 of all the farm crops are harvested? in a sample of 200 one-hectare farm lands, 96 harvested palay.
Answers
For the sample size of 200 and sample data 96 there is no sufficient evidence to conclude that proportion of palay harvested is different from 0.50.
Sample size 'n' = 200
The proportion of palay harvested is different from 0.50
Use a hypothesis test.
Let us assume the null hypothesis H₀ is that the proportion of palay harvested is equal to 0.50,
and the alternative hypothesis Hₐ is that the proportion of palay harvested is different from 0.50.
H₀: p = 0.50 proportion of palay harvested is equal to 50%
Hₐ: p ≠ 0.50 proportion of palay harvested is not equal to 50%
where p is the population proportion of palay harvested.
To test this hypothesis, use the sample data of 96 out of 200 one-hectare farm lands that harvested palay.
The sample proportion of palay harvested is,
p₁= 96/200
= 0.48
To determine if this sample proportion is significantly different from the hypothesized proportion of 0.50,
Use a two-tailed z-test with a significance level of α = 0.05.
The test statistic is calculated as,
z = (p₁ - p) / √(p(1-p)/n)
where n is the sample size.
Substituting the values, we get,
z = (0.48 - 0.50) / √(0.50(1-0.50)/200)
⇒ z = -0.5658
Using a z-table,
The probability of getting a z-value of -0.5658 or lower in the left tail of the distribution is approximately 0.7123.
Since this is a two-tailed test,
Probability of getting a z-value of 0.5658 or higher in the right tail of the distribution is also approximately 0.7123
p-value for this test is 0.7123+ 0.7123 = 1.4246
Since the p-value is greater than the significance level of α = 0.05,
Fail to reject the null hypothesis.
Therefore, do not have sufficient evidence to conclude that the proportion of palay harvested is different from 0.50.
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Simon is filling a cylindrical water dispenser that has a radius of 7 inches and a height of 20 inches. Which of these is the best estimate of the volume of this water dispenser?
A 140 in
b 2,940 in
c 840 in
d 11,769 in
PLEASE ANSWER FAST
Answers
The best estimate of the volume of this cylindrical water dispenser is 2940 in³. The correct option is b.
To estimate the volume of the cylindrical water dispenser, we can use the formula for the volume of a cylinder, which is given by V = πr²h, where V is the volume, π is a mathematical constant approximately equal to 3.14, r is the radius, and h is the height.
Given that the radius of the water dispenser is 7 inches and the height is 20 inches, we can substitute these values into the formula:
V = π(7²)(20)
V = π(49)(20)
V ≈ 3.14 × 49 × 20
V ≈ 3.14 × 980
V ≈ 3075.2 in³
From the given options, the closest estimate to the calculated volume of approximately 3075.2 in³ is option B: 2940 in³. While it is not an exact match, it is the closest estimate among the options provided.
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To the nearest hundredth, what is the value of x?
Use a trigonometric ratio to compute a distance
Answers
Therefore, to the nearest hundredth, the value of x is 42.31 units.
What is triangle?A triangle is a three-sided polygon, which is a closed shape made up of straight lines. It is one of the simplest geometric shapes and is used extensively in mathematics, science, and engineering. In a triangle, each side connects two vertices or corners, and each vertex is where two sides intersect. The three angles of a triangle always add up to 180 degrees, and the sum of the lengths of any two sides is always greater than the length of the third side. Triangles can be classified by the lengths of their sides and the sizes of their angles, which gives rise to different types such as equilateral, isosceles, scalene, acute, right, and obtuse triangles. Triangles have many applications, such as in geometry, trigonometry, physics, and engineering, and they are fundamental to understanding the properties of other shapes and mathematical concepts.
Here,
In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, for this triangle, we have:
sin(53°) = opposite / hypotenuse
sin(53°) = x / 53
To solve for x, we can rearrange the equation as follows:
x = 53 * sin(53°)
Using a calculator to evaluate sin(53°), we get:
sin(53°) = 0.7986 (rounded to four decimal places)
Substituting this value into the equation, we get:
x = 53 * 0.7986
x = 42.308 (rounded to two decimal places)
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Identify which type of sampling is used: random, stratified, cluster, systematic, or convenience.
1. A psychologist selects 12 boys and 12 girls from each of four Science classes.
2. When he made an important announcement, he based his conclusion on 10 000 responses, from
100 000 questionnaires distributed to students.
3. A biologist surveys all students from each of 15 randomly selected classes.
4. The game show organizer writes the name of each contestant on a separate card, shuffles the cards, and
draws five names.
5. Family Planning polls 1 000 men and 1 000 women about their views concerning the use of contraceptives.
6. A hospital researcher interviews all diabetic patients in each of ten randomly selected hospitals.
Answers
1. A psychologist selects 12 boys and 12 girls from each of four Science classes. = Stratified sampling
2. When he made an important announcement, he based his conclusion on 10 000 responses, from 100 000 questionnaires distributed to students= Convenience sampling
3. A biologist surveys all students from each of 15 randomly selected classes = Cluster sampling
4. The game show organizer writes the name of each contestant on a separate card, shuffles the cards, and draws five names= Random sampling
5. Family Planning polls 1 000 men and 1 000 women about their views concerning the use of contraceptives= Stratified sampling
6. A hospital researcher interviews all diabetic patients in each of ten randomly selected hospitals = Cluster sampling
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A family has four children. If Y is a random variable that pertains to the number of female children. What are the possible values of Y?
Answers
The possible values of Y are 0, 1, 2, 3, and 4.
What values can Y, the random variable for the number of female children in a family of four children, take?The number of female children in a family with four children can be any value between 0 and 4, inclusive.
To see why, we can consider all the possible outcomes of the family having four children, assuming that the probability of having a boy or a girl is 0.5 (assuming a binomial distribution).
There are 2 possibilities for the first child (boy or girl), 2 possibilities for the second child, 2 possibilities for the third child, and 2 possibilities for the fourth child, making a total of 2x2x2x2 = 16 possible outcomes.
Out of these 16 outcomes, we can count the number of outcomes that correspond to each possible value of Y:
If Y = 0, then all four children must be boys, which is 1 outcome.
If Y = 1, then there are 4 ways to have one girl (first, second, third, or fourth child).
If Y = 2, then there are 6 ways to have two girls (first two, first three, first four, second three, second four, or third fourth child).
If Y = 3, then there are 4 ways to have three girls (first three, first four, second four, or third four child).
If Y = 4, then all four children must be girls, which is 1 outcome.
Therefore, the possible values of Y are 0, 1, 2, 3, and 4.
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Which function is graphed to the right?
A. f(x) = ¹₂ +3 x-2
B. f(x)=¹+2
C. f(x) =3x+2
Answers
The rational function graphed in this problem is given as follows:
B. f(x) = 1/(x - 3) + 2.
How to obtain the rational function?From the graph, the asymptotes of the rational function are given as follows:
Vertical asymptote at x = 3 -> the function is not defined at x = 3.Horizontal asymptote at y = 2 -> as x goes to infinity, f(x) approaches y = 2.Considering that the function has a vertical asymptote at x = 3, we have that:
f(x) = 1/(x - 3).
Considering the horizontal asymptote at y = 2, we have that:
f(x) = 1/(x - 3) + 2.
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18. A singer sells a single as a music download and CD, making a total profit of £246. 64.
She sells 456 CD singles, earning 35p for every single sold.
She earns 17p for each music download of the single.
How many music downloads did she sell?
Answers
The singer sold approximately 512 music downloads for 17p and 456 CDs for 35p and earned a total profit of £246. 64.
Given data:
Total profit = £246. 64
The price for each music download is = 17p
Number of CDs sold = 456 CD
Price for each CD = 35p
Assume that the singer sold x music downloads. when she earns 17p for each music download then her total earnings from music downloads will be 0.17x. The total earnings from selling the CD are,
= 0.35 × 456
= 159.6.
From the above data, we can write the equation to find the value of x by,
0.17x + 159.6 = 246.64
0.17x = 246.64 - 159.6
0.17x = 87.04
x = 87.04/0.17
x = 512
Therefore, the singer sold approximately 512 music downloads.
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A ramp is used to go up one step.
The ramp is 3 m long. The step is 30 cm high.
How far away from the step (x) does the ramp start?
Give your answer in metres, to the nearest centimetre.
Answers
Answer:
3 meters = 300 centimeters
Using the Pythagorean Theorem:
[tex] {x}^{2} + {30}^{2} = {300}^{2} [/tex]
[tex] {x}^{2} + 900 = 90000 [/tex]
[tex] {x}^{2} = 89100[/tex]
[tex]x = 90 \sqrt{11} = 298.49[/tex]
x = about 298 centimeters
= about 2.98 meters
sets x y and z are defined below a number will be randomly selected from set x what is the probability that the selected number will be an element of set y and an element of set z.
x= (1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Y= (5, 10, 15, 20, 25)
Z= (1,5,25)
A. 0.1
B. 0.2
C. 0.5
D. 0.6
E. 0.8
Answers
When a number is chosen from set x, there is a 0.2 chance that the chosen number will also be found in sets y and z. The answer is option (B). 0.2.
What is Probability?The ratio of favourable outcomes to all possible outcomes of an event is known as the probability. The symbol x can be used to express the quantity of successful outcomes for a study with 'n' outcomes. The probability formula determines the likelihood that an event will occur.
It is the ratio of effective results to all effective results. The study of probability is a branch of mathematics that examines the likelihood that an event will occur. Probability, which expresses the likelihood that an event will occur, is calculated by dividing the total number of occurrences by the total number of positive events.
The number of elements of Both Y and Z answer is 5 and 25.
The probability of selecting a number from x that is an element of y and z
Intersection of y and z is (5, 25)
No.of elements in x is = 2/10
= 1/5
= 0.2
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By using integration by parts, find the integral 2∫⁷ in x dx b) Hence, find 2∫⁷ in √x dx
Answers
The integral is:
[tex](4/3)x^(3/2) ln(x) - (2/3)∫x^(1/2) dx = (4/3)x^(3/2) ln(x) - (4/5)x^(5/2) + C[/tex]
Solve the integrals using integration by parts.
a) To find [tex]2∫x⁷ln(x) dx[/tex], we'll use integration by parts with the formula: [tex]∫u dv = uv - ∫v du. Let's choose:u = ln(x) = > du = (1/x) dxdv = x⁷ dx = > v = (1/8)x⁸[/tex]
Now, apply the integration by parts formula:
[tex]2∫x⁷ln(x) dx = 2[uv - ∫v du] = 2[((1/8)x⁸ ln(x) - ∫(1/8)x⁸(1/x) dx)]= (1/4)x⁸ ln(x) - (1/4)∫x⁷ dx = (1/4)x⁸ ln(x) - (1/32)x⁸ + C[/tex]
b) To find 2∫√x ln(x) dx, we'll use a similar approach. Let's choose:
[tex]u = ln(x) = > du = (1/x) dxdv = √x dx = > v = (2/3)x^(3/2)[/tex]
Now, apply the integration by parts formula:
[tex]2∫√x ln(x) dx = 2[uv - ∫v du] = 2[((2/3)x^(3/2) ln(x) - ∫(2/3)x^(3/2)(1/x) dx)]= (4/3)x^(3/2) ln(x) - (2/3)∫x^(1/2) dx = (4/3)x^(3/2) ln(x) - (4/5)x^(5/2) + C[/tex]
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