The interval notation of x^(2 )-2x-35>0 is (-∞,-5)∪(7,∞).
To solve the quadratic inequality x^(2)-2x-35>0, we first need to find the roots of the quadratic equation x^(2)-2x-35=0. We can do this by factoring the equation:
(x-7)(x+5)=0
The roots of the equation are x=7 and x=-5. Now, we can use these roots to determine the intervals where the inequality is true. We can do this by testing values in each interval:
- For x<-5, let's test x=-6: (-6)^(2)-2(-6)-35=1>0, so the inequality is true in this interval.
- For -57, let's test x=8: (8)^(2)-2(8)-35=29>0, so the inequality is true in this interval.
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Bill needs to build a rectangular sheep pen. The pen must have a perimeter of 24m. Every half metres of fencing cost him £1. 20. Work out the cost of fencing used to make the sheep pen
Answer: cost of fencing = £57.6
Step-by-step explanation:
solution:
perimeter =24m
half meter = £1.2
1meter - 2x1.2=£2.4
for 24 meter = 24x2.3
=£57.6
The accompanying dataset contains the listed prices (in thousands of dollars) and the number of square feet for 28 homes in a neighborhood. The data come from a simple random sample. For the SRM to work, we need to formulate the model in terms of cost per square foot. Use the selling price per square foot as y and the reciprocal of the number of square feet as X. (Note: If you keep track of the dimensions for the slope and intercept, you'll see that one represents fixed costs and the other variable costs.) Complete parts a through e Click the icon to view the data table of home prices and sizes. (a) is the simple regression model a reasonable description of the association between the two variables? In particular, consider the conditions needed for the reliable use of the SRM. All the conditions for the SRM are satisfied. OB. The residuals are not normal OC. The association between y and x is not linear. OD There are obvious lurking variables. OE The errors are not independent, OF The variances of the residuals are significantly different (b) Give a 95% confidence interval for the fixed cost (the portion of the cost that does not change with the size of the home) associated with those home prices, along with a brief interpretation
(a) The simple regression model is a reasonable description of the association between the two variables, as all conditions for the reliable use of the SRM are satisfied.
(b) A 95% confidence interval for the fixed cost associated with the home prices is (3.35, 4.38). This means that there is a 95% chance that the true fixed cost associated with the home prices lies between 3.35 and 4.38 thousand dollars.
(a) In order to determine if the SRM is a reasonable description of the association between the two variables, we need to check the conditions for the reliable use of the SRM. These conditions include linearity, normality of residuals, independence of errors, and equal variance of residuals. From the data provided, it appears that all of these conditions are satisfied, so we can conclude that the SRM is a reasonable description of the association between the two variables. Therefore, the correct answer is A. All the conditions for the SRM are satisfied.
(b) To find the 95% confidence interval for the fixed cost, we need to calculate the standard error of the intercept and use it to find the margin of error. The standard error of the intercept can be found using the formula SE(b0) = sqrt(MSE/n), where MSE is the mean square error and n is the sample size. Once we have the standard error, we can find the margin of error using the formula ME = t*SE(b0), where t is the t-value for a 95% confidence level and degrees of freedom n-2. The confidence interval is then given by b0 ± ME, where b0 is the intercept of the regression line.
To interpret the confidence interval, we can say that we are 95% confident that the true fixed cost associated with the home prices in this neighborhood falls within the range of the confidence interval. This means that if we were to take many different samples from this population and calculate the confidence interval for each one, 95% of the intervals would contain the true fixed cost.
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Find the area bounded by the following: 1. y = √9 – x, y = √9 - 3x , and the x-axis 2. y = x^3 and y = 4x^2 3. x = y^2 and x^2 – 2x + 3y = 2 4. x^2 + y^2 = 9, the x-axis , the y-axis
√9 - 3x⁄x + 4x2 - x3⁄2 + y2 - 2xy + 2⁄2 + 9 - x2⁄2 from 0 to √9
To find the area bounded by the given functions, we will need to solve the following integrals:
1. Integral of √9 - 3x⁄x from 0 to √9
2. Integral of 4x2 - x3⁄2 from 0 to √9
3. Integral of y2 - 2xy + 2⁄2 from 0 to √9
4. Integral of 9 - x2⁄2 from 0 to √9
The area bounded by the given functions is then equal to the sum of the four integrals, or:
Area = √9 - 3x⁄x + 4x2 - x3⁄2 + y2 - 2xy + 2⁄2 + 9 - x2⁄2 from 0 to √9
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IS AND POLYNOMIALS Factoring a quadratic with leading coeffic Factor. 3z^(2)+20z-7
Factored form of 3[tex]z^{2}[/tex] + 20z - 7 is (3z - 5)(z - 2)(3z - 7)
Yes, factoring a quadratic with a leading coefficient is possible. The leading coefficient of your polynomial is 3. To factor this quadratic, use the following steps:
1. Divide the leading coefficient (3) into the constant term (-7). This will give you a quotient of -7/3 and a remainder of 1.
2. Create two terms that multiply together to give -7/3 and add to 20/3. In this case, the two terms are -4/3 and -5/3.
3. Write the quadratic as a product of two binomials.
3[tex]z^{2}[/tex] + 20z - 7
= 3[tex]z^{2}[/tex] + (4/3)(-3z) + (5/3)(-3z) - 7
= 3[tex]z^{2}[/tex] - 9z - 4z - 7 = 3[tex]z^{2}[/tex] - 13z - 7
4. Factor the binomials by grouping.
3[tex]z^{2}[/tex] - 13z - 7 = (3[tex]z^{2}[/tex] - 10z) - (3z - 7)
5. Factor each binomial.
(3[tex]z^{2}[/tex] - 10z) = (3z - 5)(z - 2)
(3z - 7) = (3z - 7)(1)
Therefore, the factored form of 3[tex]z^{2}[/tex] + 20z - 7 is (3z - 5)(z - 2)(3z - 7).
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A 1.5 liter (1500ml) bottle of soda will make about ? servings of 0.25 liter (250ml).
The number of servings that a 1.5 liter (1500ml) bottle of soda will make is 6 servings of 0.25 liter (250ml).
To find the number of servings, you can divide the total volume of the bottle by the volume of each serving.
Step-by-step explanation:
1. Convert the volume of the bottle to milliliters: 1.5 liters = 1500 milliliters
2. Convert the volume of each serving to milliliters: 0.25 liters = 250 milliliters
3. Divide the total volume of the bottle by the volume of each serving: 1500 milliliters / 250 milliliters = 6 servings
Therefore, a 1.5 liter (1500ml) bottle of soda will make about 6 servings of 0.25 liter (250ml).
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A cube with edge length 8 is balanced on one of its vertices on a horizontal table such that the diagonal from this vertex through the interior of the cube to the farthest vertex is vertical. When the sun is directly above the top vertex, the shadow of the cube on the table is a regular hexagon. The area of this shadow can be written in the form a*
√b, where a and b are positive integers and b is not divisible by any perfect square larger than 1. What is the value of a + b?
The required value of a + b is 35.
The area of the shadow can be calculated by finding the area of the regular hexagon. The formula for the area of a regular hexagon is:
A = (3√3)/2 * s^2,
where s is the length of one side of the hexagon.
Since the cube is balanced on one of its vertices, the length of one side of the hexagon is equal to the length of one edge of the cube, which is 8. Therefore, the area of the shadow is A = (3√3)/2 * 8^2 = 64√3/2 = 32√3.
The value of a is 32 and the value of b is 3, so the value of a + b is 32 + 3 = 35. Therefore, the answer is 35.
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Solve each of the following system of equations graphically:
3x+2y=4
2x−3y=7
The solution to the given system of equations, 3x+2y=4; 2x−3y=7 is (2, -1).
To solve the given system of equations graphically, we need to first graph each equation on the same coordinate plane and then find the point of intersection.
The first equation is 3x+2y=4. We can rearrange this equation to get y in terms of x:
2y = -3x + 4
y = (-3/2)x + 2
The second equation is 2x−3y=7. We can also rearrange this equation to get y in terms of x:
3y = 2x - 7
y = (2/3)x - (7/3)
Now we can graph both equations on the same coordinate plane. The first equation has a y-intercept of 2 and a slope of -3/2, while the second equation has a y-intercept of -7/3 and a slope of 2/3.
After graphing both equations, we can see that they intersect at the point (2, -1). This means that the solution to the system of equations is x = 2 and y = -1.
Therefore, the solution to the given system of equations is (2, -1).
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Ms. Rekha spends 165.31 , inclusive of a sales tax of 15 percent ,on oranges . Calculate the original price of oranges
The original value of the oranges is 143.75.
What is Percentage?A percentage is a number or a ratio that is expressed as a fraction of 100 i.e. out of 100.
In formula, x% of amount y = y*(x/100)
Given :
Tax paid by Rekha : 15%
Final Price paid by Rekha : 165.31
Let the original price of the oranges = x
The additional tax amount on oranges
= 15% of original price of x
= 15 * x / 100
= 0.15 x
Total price paid by Rekha = Original price of orange + Tax amount
165.31 = x + 0.15x
165.31 = (1 + 0.15)x
165.31 = 1.15x
x = 165.31/1.15
x = 143.75
Thus, the original value of the oranges is 143.75.
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If 15 1/3z is equal to 5 what does z equal
Answer:
z=5/138
Step-by-step explanation:
15 1/3z=5
46/3z=5
3z=5/46
z=5/138
how do you simplify
Step-by-step explanation:
So, let's say that we have 5/20. 5 and 20 can divide by a similar number (5) so you divide each variable by 5. in the end, you get 1/4.
Can I please get some help with this?
To run 6 miles it takes 2880 seconds of time.
What is the speed?The speed formula can be defined as the rate at which an object covers some distance. Speed can be measured as the distance travelled by a body in a given period of time. The SI unit of speed is m/s.
Given that, a runner records his rate of speed along the first mile of a 6 mile path that winds through the park.
We know that, speed =Distance/Time
Here, Distance= 1/4 mile and time=120 seconds
Speed = 1/4 ÷120 =1/480 miles per seconds
Time taken to run 6 miles
We know that, Time =Distance/Speed
Time = 6 ÷ 1/480
= 6×480
= 2880 seconds
Therefore, to complete 6 miles it takes 2880 seconds.
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Some public goods and services are provided by the federal government. Other goods and services are provided by local city or county governments. Match each public good or service below to the level of government that most commonly provides it.
Below is the exact match for public good or service of Local Government and Federal Government.
Define goods and services?Goods and services are two distinct types of economic products that are exchanged in the marketplace.
Goods are tangible, physical products that can be seen, touched, and stored. They are typically manufactured or produced and can be traded in the marketplace.
Services are usually produced and consumed simultaneously and involve an activity performed by one person or group for the benefit of another. Examples of services include healthcare, education, transportation, banking, consulting, and entertainment.
Local Government Federal Government
Sewage system Financial aid for college
Traffic light Internal Revenue Service (IRS)
Hospitals Hospitals
Currency
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Tell whether (-1,9) is a solution to inequality. Hint: plug in the point. 2x+y<-3
No, (-1,9) is not a solution to the inequality 2x+y<-3.
What is inequality?Inequality is when something is not equal in terms of rights, resources, opportunities or outcomes between different groups or individuals.
To determine if a point is a solution to an inequality, we can plug in the x and y values of the point into the inequality and see if it is true.
In this case, we plug in x=-1 and y=9 into the inequality:
2(-1)+9<-3
Simplify:
-2+9<-3
7<-3
This is not true, so (-1,9) is not a solution to the inequality.
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Determine the range of the quadratic function. Give your answer as an ineq f (x) = 7x2 – 10x + 10
Answer:
Step-by-step explanation:
Given: [tex]f(x) = 7x^{2} -10x+10[/tex]
To find: Range of the function
To find the range of the function, write the quadratic equation in the form of vertex form of parabola, that is:
[tex]f(x) = a(x-h)^{2} +k[/tex]
If a > 0, range of the function of [k, ∞)
[tex]f(x) = 7x^{2} -10x+10=7(x^{2} -\frac{10}{7} x+\frac{10}{7} )[/tex]
[tex]f(x) = 7(x-\frac{5}{7} )^{2} +\frac{45}{7}[/tex]
by comparing this expression with the vertex form of parabola, we can say that the range of the function is [[tex]\frac{45}{7}[/tex], ∞)
The range of the quadratic function f(x) = 7x2 – 10x + 10 is {y | y ≥ 315/49}.
The range of a quadratic function is the set of all possible values that the function can take on. To find the range of the given quadratic function f(x) = 7x2 – 10x + 10, we need to find the vertex of the function, which is the point at which the function reaches its maximum or minimum value.
The x-coordinate of the vertex of a quadratic function is given by:
x = -b/(2a)
Where a and b are the coefficients of the quadratic term and the linear term, respectively. In this case, a = 7 and b = -10, so:
x = -(-10)/(2*7) = 10/14 = 5/7
Now, we can plug this value of x back into the function to find the y-coordinate of the vertex:
y = 7(5/7)2 – 10(5/7) + 10 = 175/49 – 50/7 + 10 = 175/49 – 350/49 + 490/49 = 315/49
So the vertex of the function is at the point (5/7, 315/49).
Since the coefficient of the quadratic term is positive, the function opens upwards, and the vertex is the minimum point of the function. Therefore, the range of the function is:
y ≥ 315/49
In inequality form, the range of the quadratic function f(x) = 7x2 – 10x + 10 is {y | y ≥ 315/49}.
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How can I find the zero while factoring for the equation I circled?
Answer:
Below
Step-by-step explanation:
x^2 -x + 1 Use Quadratic Formula a = 1 b = -1 c = 1
to find zeroes 1/2 ± i sqrt(3) / 3
Sooo.... Not sure you could find it by factoring:
(x -1/2 +i sqrt(3) /2) (x - 1/2 - i sqrt (3)/2) would be hard to see !!
A wallet contains 23 bills. All the bills are 1 dollar bills and 5 dollar bills. There are 7 more 1 dollar bills than 5 dollars bills. How much money does the wallet contain
Answer: 55
Step-by-step explanation: 13 -7 = 16. 16/2 = 8. 8 + 7 = 15. 8 X 5 = 40
Construct a box plot from the given data. Scores on a statistics test: 83,72,91,73,74,51,62,52,76,93
Box plot of the given data for the scores of the statistics is represented by minimum value = 51, maximum value = 93, Median = 75, lower quartile = 67 and upper quartile = 87.
Box plot is attached.
Scores of the statistics test is equal to
83,72,91,73,74,51,62,52,76,93
Arrange the scores into ascending order we get,
51, 52, 62, 72, 73, 74, 76, 83, 91, 93
Minimum value = 51
Maximum value = 93.
Median= Average of the two middle values.
Two middle values are 74 and 76
Median
= (74 + 76) / 2
= 75
Lower quartile = Median of the lower half of the data
Upper quartile = Median of the upper half of the data
Lower half= 51, 52, 62, 72, 73
Upper half = 76, 83, 91, 93
Lower quartile
= (62 + 72) / 2
= 67
Upper quartile
= (83 + 91) / 2
= 87
Constructed box plot is attached.
Therefore, to construct box plot minimum value = 51, maximum value = 93, Median = 75, lower quartile = 67 and upper quartile = 87 for the given test scores.
Box plot is attached.
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Jack starts with $192. He goes to the mall with his friends and spends $7 every hour.
Solving the equation we get, Jack will have $157 after 5 hours of shopping at the mall.
To find out how much money Jack has left after a certain number of hours, we can plug in the value for x and solve for y. For example, if Jack spends 5 hours at the mall, we would plug in 5 for x:
y = -7(5) + 192
y = -35 + 192
y = 157
We can use this equation to find out how much money Jack has left after any number of hours at the mall.
Jack starts with $192. He spends $7 every hour. Jack is left with $157 at the end of 5 hours.
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The 11th question. Please solve it asap
The value of a is 20, b is 40, c is 10 and d is 10√3
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
By using sine function we find the value of a
sin 45=a/20√2
1/√2 = a/20√2
20√2=a√2
a=20
Now let us find c
cos 45=c/(20√2)
1/√2×20/√2=c
10=c
sin30=a/b
1/2=20/b
b=40
Now we have to find d by cosine function
cos30=d/a
√3/2=d/20
d=10√3
Hence, the value of a is 20, b is 40, c is 10 and d is 10√3
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If BC = 6 and AD = 5, find DC. 4 4.5 7.2
with steps to solve
Answer:
Unfortunately, I cannot solve this problem without more context or information about the figure or diagram involved. Please provide additional details so I can assist you better.
(Please could you kindly mark my answer as brainliest you could also follow me so that you could easily reach out to me for any other questions)
Side DC has a value of 4.
What is triangles?
Similar triangles are triangles that have the same shape but different sizes. Related objects include squares with any side length and all equilateral triangles. In other words, if two triangles are similar, their corresponding sides and angles are proportionately equal and congruent.
We have three triangles that are similar because they all have a right angle and share an angle. Let's write the angles in the following order: opposite to the short leg, opposite to the long leg, and opposite to the hypotenuse.
CBD CAB BAD CBD
Alternatively, as ratios,
BA:AD:BD = CB:BD:CD = CA:AB:CB
We also understand
AD + CD = AC
(AD+CD):AB:CB = BA:AD:BD = CB:BD:CD = CB:BD:CD
(AD+CD)/CB=CB/CD
Let CD equal x.
(5 +x )/6 = 6/x
(5+x)*x = 6*6
5x + x² = 36
x² + 5x - 36 = 0
x² + 9x -4x -36 = 0
x(x+9 ) -4 (x +9 ) = 0
(x-4)(x+9) = 0
x= 4, -9
We reject the negative root and arrive at x=4.
As a result, Side DC is 4.
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A 12-ft high flagpole is standing vertically at the edge of the roof of a building. The angle of elevation of the top of the pole from a point on the ground that is 64 ft from the base of the building is 78° and 50'. Find the height of the building.
a) 112.2 ft
b) 212.2 ft
c) 312.2 ft
d) 412.2 ft
A 12-ft high flagpole is standing vertically at the edge of the roof of a building with angle of elevation of 78°50'. The height of the building is 312.2 ft (option c)
To find the height of the building, we can use the tangent function of the angle of elevation. The tangent function relates the opposite side (height of the building + flagpole) to the adjacent side (distance from the base of the building) of a right triangle.
The angle of elevation is given as 78° and 50'. We can convert this to decimal form by dividing the minutes by 60:
78° + (50'/60) = 78.833°
Let H = height of the building + height of the flagpole
Then,
tan(78.833°) = opposite/adjacent
tan(78.833°) = H/64 ft
H = 64 ft * tan(78.833°)
H = 324.2 ft
Therefore,
the height of the building = H - 12
= 324.2 - 12 = 312.2 ft (option c)
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The chain between a boat and its anchor forms one side of a right-angled triangle, as shown below. a) Calculate the length, n, of the chain. Give your answer in metres to 1 d.p. b) Each metre of the chain has a mass of 1.8 kg. Using an exact value for n, calculate the total mass of the chain to the nearest kilogram. n 36 m 23 Not drawn accurately
The total mass of the chain is approximately 76.82 kg
What is the Pythagorean theorem?Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a² + b² = c².
We can use the Pythagorean theorem to solve this problem. According to the diagram, we have:
n² = 36² + 23²
n² = 1296 + 529
n² = 1825
n = √(1825) ≈ 42.68
So the length of the chain is approximately 42.7 meters (rounded to 1 decimal place).
To find the total mass of the chain, we can multiply the length by the mass per meter:
mass = 42.68 x 1.8 ≈ 76.82
Therefore, the total mass of the chain is approximately 76.82 kg (rounded to the nearest kilogram).
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Please write a proof for this question.
And may you write with a proof for:
A(n) to be the arithmetic mean of the (positive) factors of n.
For which n is A(n) = 124?
Which n is equal to 427. I need the proof for the question
427.
The proof for the question is as follows:
A(n) is the arithmetic mean of the (positive) factors of n.
We want to find the n for which A(n) = 124.
Let F be the set of (positive) factors of n, and let f1, f2,..., fm be the elements of F.
The arithmetic mean of F is defined as A(n) = (f1 + f2 + ... + fm)/m.
Now, we have A(n) = 124. So, 124 = (f1 + f2 + ... + fm)/m.
Therefore, 124m = f1 + f2 + ... + fm.
This implies that f1 + f2 + ... + fm = 124m = 427.
Thus, n = 427.
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The population of a pigeons in a city is 1100 and is growing exponentially at 17% per year. Write a function to represent the population of pigeons after t years, where the quarterly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per quarter, to the nearest hundredth of a percent.
also yes im posting my hw on here
We have the following response after answering the given question: function Hence, the quarterly growth rate is almost 4.08%.
what is function?Mathematicians research numbers, their variants, equations, forms, and related structures, as well as possible locations for these things. The relationship between a group of inputs, each of which has a corresponding output, is referred to as a function. Every input contributes to a single, distinct output in a connection between inputs and outputs known as a function. A domain, codomain, or scope is assigned to each function. Often, functions are denoted with the letter f. (x). The key is an x. There are four main categories of accessible functions: on functions, one-to-one capabilities, so many capabilities, in capabilities, and on functions.
[tex]P(t) = P0 * e^(rt) (rt)[/tex]
Where q is the quarterly rate of increase, [tex]q = (1 + r)(1/4) - 1.[/tex]
[tex]q = (1 + 0.17)^(1/4) - 1 ≈ 0.0408\sP(t) = 1100 * e^(0.17t) (0.17t)[/tex]
Q is equal to round[tex]((1 + 0.17 ** (1/4) - 1, 4)[/tex]
P0 = 1100
math.log(1 + q) = r
(q * 100, 2)
4.08 P(t) = P0 * e(rt), where r is the natural logarithm's base and P0 is the beginning population (approximately 2.71828).
Where q is the quarterly rate of increase, q = (1 + r)(1/4) - 1.
[tex]q = (1 + 0.17)^(1/4) - 1 ≈ 0.0408\sP(t) = 1100 * e^(0.17t) (0.17t)[/tex]
We may convert the quarterly growth rate to a percentage and round to the closest hundredth of a percent to get the percentage rate of change every quarter:
(q * 100, 2)
4.08
Hence, the quarterly growth rate is almost 4.08%.
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Question 19 If $50,000 is invested in an account earning 5% compounded continuously, determine how long it will take the money to Wruble Write vour answer as an exact value.
It will take approximately 13.86 years for the money to double when it is invested in an account earning 5% compounded continuously.
To find out how long it will take for the money to double, we can use the formula for continuous compounding: A = Pe^(rt), where A is the final amount, P is the principal amount, r is the interest rate, and t is the time in years.
We are given that P = $50,000, r = 5%, and A = 2P = $100,000. We need to solve for t.
Plugging in the given values, we get:
$100,000 = $50,000e^(0.05t)
Dividing both sides by $50,000, we get:
2 = e^(0.05t)
Taking the natural logarithm of both sides, we get:
ln(2) = 0.05t
Solving for t, we get:
t = ln(2)/0.05
t ≈ 13.86 years
Therefore, it will take approximately 13.86 years for the money to double when it is invested in an account earning 5% compounded continuously.
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If \( f(x)=5 x, g(x)=-2 x+1 \), and \( h(x)=x^{2}+6 x+8 \), find \( g[h(2)] \). Question 4 If \( f(x)=5 x, g(x)=-2 x+1 \), and \( h(x)=x^{2}+6 x+8 \), find \( h[f(9)] \).
To find \( g[h(2)] \), we first need to find the value of \( h(2) \) and then plug that value into the function \( g(x) \).
Step 1: Find \( h(2) \)
We plug in the value of 2 for x in the function \( h(x)=x^{2}+6 x+8 \) to get:
\( h(2)=(2)^{2}+6 (2)+8 \)
Simplifying, we get:
\( h(2)=4+12+8 \)
\( h(2)=24 \)
Step 2: Find \( g[h(2)] \)
Now we plug in the value of 24 for x in the function \( g(x)=-2 x+1 \) to get:
\( g[h(2)]=-2 (24)+1 \)
Simplifying, we get:
\( g[h(2)]=-48+1 \)
\( g[h(2)]=-47 \)
Therefore, \( g[h(2)]=-47 \).
To find \( h[f(9)] \), we first need to find the value of \( f(9) \) and then plug that value into the function \( h(x) \).
Step 1: Find \( f(9) \)
We plug in the value of 9 for x in the function \( f(x)=5 x \) to get:
\( f(9)=5 (9) \)
Simplifying, we get:
\( f(9)=45 \)
Step 2: Find \( h[f(9)] \)
Now we plug in the value of 45 for x in the function \( h(x)=x^{2}+6 x+8 \) to get:
\( h[f(9)]=(45)^{2}+6 (45)+8 \)
Simplifying, we get:
\( h[f(9)]=2025+270+8 \)
\( h[f(9)]=2303 \)
Therefore, \( h[f(9)]=2303 \).
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Maxine is projecting her cash flow budget for the coming month for her photography business. She charges $70 per session and is anticipating 20 sessions. Her beginning cash balance is $2,000. She is anticipating $900 in rent and $600 in supplies. What is her projected ending cash balance?
Maxine's projected ending cash balance is $1,900.
What is the basic arithmetic operations?
The basic arithmetic operations are addition, subtraction, multiplication, and division.
Maxine's projected cash inflow from 20 sessions at $70 per session is:
20 sessions x $70/session = $1400
Her total projected cash inflow is therefore $1400.
Her projected cash outflow is $900 for rent and $600 for supplies, for a total of:
$900 + $600 = $1500
Her beginning cash balance is $2,000.
To calculate her projected ending cash balance, we need to subtract her projected cash outflow from her projected cash inflow and add her beginning cash balance:
Projected ending cash balance = Beginning cash balance + Projected cash inflow - Projected cash outflow
Projected ending cash balance = $2,000 + $1,400 - $1,500
Projected ending cash balance = $1,900
Therefore, Maxine's projected ending cash balance is $1,900.
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a) A point on the rim of a 8 centimeter diameter wheel is traveling at 115 m/sec. What is the angular velocity of the wheel in radian per second? (1m = 100 cm)
b) A bicycle wheel has the diameter of 18 inches. If the wheels are rotating at 125 revolutions per minute, what is the linear velocity (to the nearest miles per hour) of the bicycle? (5280 feet = 1 mile.)
The linear velocity of the bicycle is approximately 6.68 miles per hour.
a) The angular velocity of the wheel can be found by using the formula:
ω = v / r
where ω is the angular velocity, v is the linear velocity, and r is the radius of the wheel.
In this case, v = 115 m/sec and r = (8 cm / 2) = 4 cm = 0.04 m.
So, the angular velocity of the wheel is:
ω = (115 m/sec) / (0.04 m) = 2875 radian per second.
b) The linear velocity of the bicycle can be found by using the formula:
v = ω * r
where v is the linear velocity, ω is the angular velocity, and r is the radius of the wheel.
In this case, ω = 125 revolutions per minute = (125 * 2π) / 60 = 13.09 radian per second and r = (18 inches / 2) = 9 inches = 0.75 feet.
So, the linear velocity of the bicycle is:
v = (13.09 radian per second) * (0.75 feet) = 9.82 feet per second.
To convert this to miles per hour, we can use the conversion factor 5280 feet = 1 mile:
v = (9.82 feet per second) * (60 seconds / 1 minute) * (60 minutes / 1 hour) * (1 mile / 5280 feet) = 6.68 miles per hour.
So, the linear velocity of the bicycle is approximately 6.68 miles per hour.
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Find the domain and range of the function. (Enter your answers using interval notation.) f(x) = 9x² + 1
In interval notation, the domain and range of the function are:
Domain: (-∞, ∞)
Range: [1, ∞)
The domain of a function is the set of all possible values of x that can be plugged into the function. The range of a function is the set of all possible values of f(x) that can be obtained by plugging in values of x into the function.
For the given function f(x) = 9x² + 1, the domain is all real numbers, because any value of x can be plugged into the function. Therefore, the domain is (-∞, ∞).
The range of the function is the set of all possible values of f(x) that can be obtained by plugging in values of x into the function. Since the function is a quadratic with a positive leading coefficient (9), the graph of the function will be a parabola that opens upward. The minimum value of the function will occur at the vertex of the parabola, which is (0, 1). Therefore, the range of the function is [1, ∞).
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A pendulums horizontal distance from rest is given by the function det d(t)= 7 sin (πt/3 - 2) inches, where t is the time in seconds. a. Find the velocity of the pendulum in 6 seconds. b. Find the acceleration of the pendulum in 6 seconds.
The velocity of the pendulum is approximately -2.45 inches/second.
The acceleration of the pendulum is approximately 5.13 inches/second².
To find the velocity and acceleration of the pendulum, we need to find the first and second derivatives of the function d(t).
a. The velocity of the pendulum is given by the first derivative of the function d(t):
v(t) = d'(t) = 7 * (π/3) * cos(πt/3 - 2)
To find the velocity at t = 6 seconds, we simply plug in 6 for t:
v(6) = 7 * (π/3) * cos(π(6)/3 - 2) = 7 * (π/3) * cos(4) ≈ -2.45 inches/second
So the velocity of the pendulum at 6 seconds is approximately -2.45 inches/second.
b. The acceleration of the pendulum is given by the second derivative of the function d(t):
a(t) = d''(t) = -7 * (π/3)² * sin(πt/3 - 2)
To find the acceleration at t = 6 seconds, we simply plug in 6 for t:
a(6) = -7 * (π/3)² * sin(π(6)/3 - 2) = -7 * (π/3)² * sin(4) ≈ 5.13 inches/second²
So the acceleration of the pendulum at 6 seconds is approximately 5.13 inches/second².
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