The recursive formula for the sequence is an = an-1 + 2. The explicit formula is an = 9 + (n-1)(2). There are 31 seats in the 12th row.
Explanation:a) Recursive formula: The difference between the numbers of seats in consecutive rows is always 2. So, the recursive formula can be written as: an = an-1 + 2. Here, an represents the number of seats in the nth row, and an-1 represents the number of seats in the (n-1)th row.
b) Explicit formula: The first term of the arithmetic sequence is 9, and the common difference is 2. The explicit formula can be written as: an = a1 + (n-1)d. Substituting the values, we get an = 9 + (n-1)(2).
c) Seats in the 12th row: Using the explicit formula, we can calculate the number of seats in the 12th row: a12 = 9 + (12-1)(2) = 9 + 22 = 31. So, there are 31 seats in the 12th row.
A profit function is derived from the production cost and revenue function for a given item. The monthly profit function for a certain item is given by P(x)=−0.05x^2+500x−100,000, where P is in dollars and x is the number of units sold. How many units maximize the profit? FInd the maximum profit
Now suppose the roster has 3 guards, 5 forwards, 3 centers, and 2 "swing players" (x and y) who can play either guard or forward. if 5 of the 13 players are randomly selected, what is the probability that they constitute a legitimate starting lineup? (round your answer to three decimal places.)
The problem calculates the probability of forming a legitimate basketball starting lineup from a given roster, using combinatorial methods to determine the ratio of favorable outcomes to total outcomes.
Explanation:The question asks for the probability of selecting a legitimate starting lineup for a basketball team, given a roster with specific numbers of guards, forwards, centers, and swing players. A standard basketball starting lineup consists of 2 guards, 2 forwards, and 1 center. Given the roster has 3 guards, 5 forwards, 3 centers, and 2 swing players (who can play either guard or forward), we calculate the probability of forming a legitimate lineup through combinatorial methods.
To calculate the total number of ways to form a starting lineup, we consider the swing players as forwards when calculating combinations since they can play either role. The total number of ways to choose 2 guards out of 5 possible options (3 guards + 2 swing players), 2 forwards out of 7 possible options (5 forwards + 2 swing players treated as forwards), and 1 center out of 3 options is given by the product of combinations: C(5,2) * C(7,2) * C(3,1).
The total number of ways to select any 5 players out of the 13 (without regard for position) is C(13,5). Therefore, the probability is the ratio of these two numbers, rounded to three decimal places.
Suppose a certain population of observations is normally distributed. What percentage of the observations in the population.
(a) are within + 1.5 standard deviations of the mean?
(b) are more that 2.5 standard deviations above the mean?
(c) are more that 3.5 standard deviations away from ( above or below) the mean?
The required percentages are:
(a) 86.64%
(b) 0.62%
(c) 0.04%
With the use of standard normal table, we can find the required percentage, such as:
(a)
→ [tex]P( -1.5<z<1.5)= P( z <1.5)- p( z < -1.5)[/tex]
[tex]= 0.9332-0.0668[/tex]
[tex]= 0.8664[/tex]
[tex]= 86.64[/tex] (%)
(b)
→ [tex]P( z >2.5)=0.0062[/tex]
[tex]= 0.62[/tex] (%)
(c)
→ [tex]P( z < -3.5) + p( z > 3.5) = 0.0002+0.0002[/tex]
[tex]= 0.0004[/tex]
[tex]=0.04[/tex] (%)
Thus the above approach is right.
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Anne plans to save $40 a week for the next five years. she expects to earn 3 percent for the first two years and 5 percent for the last three years. how much will her savings be worth at the end of the five years
Anne's savings will be worth $11,636.924 at the end of the five years.
We have,
PV= $40
Future Value = Present Value (1 + Interest Rate[tex])^{Time[/tex]
For the first two years:
Present Value = $40 per week * 52 weeks/year * 2 years = $4,160
Interest Rate = 3% = 0.03
Time = 2 years
Future Value (first two years) = $4,160 (1 + 0.03)²= $ 4,413.344
For the last three years:
Present Value = $40 per week * 52 weeks/year * 3 years = $6,240
Interest Rate = 5% = 0.05
Time = 3 years
Future Value (last three years) = $6,240 (1 + 0.05)³ = $ 7,223.58
Then, Total Future Value = Future Value (first two years) + Future Value (last three years)
= 4413.344 + 7223.58
= $ 11,636.924
Therefore, Anne's savings will be worth $11,636.924 at the end of the five years.
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what is the volume of tue box pictured below 5/6, 1 1/8,4/5
Answer:
The answer is 3/4
Step-by-step explanation:
Help me please because I can't finish it
Pamela is 11 years older than Jiri. The sum of their ages is 77 . What is Jiri's age?
An aquarium 7 m long, 1 m wide, and 1 m deep is full of water. find the work needed to pump half of the water out of the aquarium. (use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.) show how to approximate the required work by a riemann sum. (let x be the height in meters below the top of the tank. enter xi* as xi.) lim n → ∞ n i = 1 δx express the work as an integral. 0 dx evaluate the integral. j
Final answer:
The work needed to pump out half the water from the given aquarium is 17,150 Joules. This is determined by applying physical principles to calculate the mass of the water, the effect of gravity, and using an integral to find the total work done against gravity.
Explanation:
The student is asking about the work required to pump half of the water out of an aquarium with dimensions 7 m long, 1 m wide, and 1 m deep using physics concepts involving work, force, and Riemann sums. To approach this problem, we must consider the work done against gravity to move the water from its initial position to the top of the aquarium. The density of water (ρ) is 1000 kg/m³, and the acceleration due to gravity (g) is 9.8 m/s².
First, calculate the volume of water to be pumped out, which is half the aquarium volume: V = ½ × 7 m × 1 m × 1 m = 3.5 m³. Convert this volume to mass using the density of water, m = ρV = 1000 kg/m³ × 3.5 m³ = 3500 kg.
The work done to pump out half the water can be calculated using the concept of the center of mass of the water being lifted, which is at a height h/2 from the top of the water when the tank is half full, where h is the depth of the tank. Therefore, the work is W = mgh/2 = 3500 kg × 9.8 m/s² × 0.5 m = 17150 J.
To approximate the required work using a Riemann sum, consider the small amount of work to lift a thin layer of water δx from a depth x to the top of the tank, dW = ρgAdx(x), where A is the area of the tank's surface. We set up the integral ∫ W = ρgA ∫ xdx from 0 to h/2, and find the limit as the number of partitions goes to infinity. The integration gives us the same work W = 17150 J.
A savings bank invests $58,800 in municipal bonds and earns 12% per year on the investment. How much money is earned per year?
The larger of two numbers is eight more than three times the smaller number.The sum of the two numbers is forty-eight.Find the two numbers.
When no unit is given for an angle, what unit must be used?
The CBS’ television show 60 Minutes has been successful for many years. That show recently had a share of 20, which means that among the TV sets in use at the time the show aired, 20% were tuned to 60 Minutes. Assume that this is based on a sample size of 5000 – which is a typical sample size for this kind of experiments. Construct a 95% confidence interval for the true proportion of TV sets that are tuned to 60 Minutes..
When no unit is given for an angle, what unit must be used?
Find three real numbers x, y, and z whose sum is 6 and the sum of whose squares is as small as possible. g
The numbers that fit the conditions of the question are x = 2, y = 2, and z = 2. This sums to 6 and minimizes the sum of their squares (12).
Explanation:The subject of this problem involves real numbers and their sums and squares. The intention is to find three real numbers (x, y, and z) such that their sum equals 6, and the sum of their squares is minimized. By symmetry, it is preferable if these three numbers are equal. Therefore, x = y = z = 6/3 = 2 is the optimal solution.
So the three real numbers are x = 2, y = 2, and z = 2, which sum to 6 and the sum of their squares is as small as possible (12).
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(a) find the position vector of a particle that has the given acceleration and the specified initial velocity and position. a(t) = 8t i + sin t j + cos 2t k, v(0) = i, r(0) = j
Final answer:
To find the position vector given the acceleration, initial velocity, and initial position, we integrate the acceleration vector to find the velocity vector, and then integrate the velocity vector to find the position vector. The resulting position vector is r(t) = (⅔t³ + t)i - sin t j - ¼ cos 2t k.
Explanation:
Finding the Position Vector
The question asks us to find the position vector of a particle given its acceleration vector a(t) = 8t i + sin t j + cos 2t k, initial velocity v(0) = i, and initial position r(0) = j. To find the position vector, we first need to integrate the acceleration vector to find the velocity vector, and then integrate the velocity vector to find the position vector.
Step 1: Find the Velocity Vector
Integrate the acceleration vector for time to get the velocity vector. The indefinite integral of the acceleration vector gives:
Vx = ½8t² + C1Vy = -cos t + C2Vz = ½sin 2t + C3Using the initial velocity v(0) = i, we find C1 = 1, C2 = 0, and C3 = 0. Therefore, the velocity vector is v(t) = (4t² + 1)i - cos t j + ½sin 2t k.
Step 2: Find the Position Vector
Integrate the velocity vector concerning time to get the position vector. The indefinite integral of the velocity vector gives:
Rx = ⅔t³ + t + C4Ry = -sin t + C5Rz = -¼ cos 2t + C6Using the initial position r(0) = j, we find C4 = 0, C5 = 1, and C6 = 0. Thus, the position vector is r(t) = (⅔t³ + t)i - sin t j - ¼ cos 2t k.
(APEX) Factor a number, variable, or expression out of the trinomial shown below:
4x2 – 16x + 8
A.2(x2 – 8x + 4)
B.4(x2 – 4x)
C.8(x2 – 2x + 1)
D.4(x2 – 4x + 2)
How will the solution of the system y>2x+2/3 and y<2x+1/3 change if the inequality sign on both inequalities is reversed to y<2x+2/3 and y>2x+1/3?
Sample Response: There is no solution to the system in its original form. There are no points in common. If the signs are reversed, the system has an intersection with an infinite number of solutions.
If f(x) = sqrt x-3, which inequality can be used to find the domain of f(x)?
sqrt x-3 >/= 0
x - 3 >/= 0
sqrt x - 3 = 0
x - 3 = 0
Answer: B: x-3 >_ 0
Step-by-step explanation:
Took test
h(t)=15-10t-16t^2 if a snowboarders horizontal velocity is 10feet per second, how far from the base of the overhang will she land? 15 equals initial height of overhang, -10 is the initial vertical velocity and t is the time.
Which transformations could be preformed to show that ABC is similar A"B"C"?
Kim scored the least number of points. Claire scored five more points than Kim. Sam scored twice as many points as Kim. Together the three students scored 85 points. How many points did each student score?
Abed says he has written a system of two linear equations that has an infinite number of solutions. One of the equations of the system is y = 3x – 1. Which could be the other equation? y = 3x + 2 3x – y = 2 3x – y = 1 3x + y = 1
Answer:
3rd Option is correct.
Step-by-step explanation:
Given: Equation is y = 3x - 1
We need to find another equation such that system of equation has infinitely solutions.
We know that System of Equations having infinitely many solution has ratio as following,
[tex]\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}[/tex]
where a is coefficient of x and b is coefficient of y while c is constant term.
Clearly from Given Options Second equation is not a line with different coefficients. Its actually same with some changes.
Consider,
y = 3x - 1
transpose 3x to LHS
y - 3x = -1
Multiply both sides with -1
-y + 3x = 1
3x - y = 1
Therefore, 3rd Option is correct.
Isaiah spent $19.60 on a gift for his mother. The amount that he spent on the gift was 5/7 of the total amount that he spent at the store. Which statements can be used to find x, the total amount that Isaiah spent at the store? Check all that apply
The total amount that Isaiah spent at the store will be equal to $27.44.
What is an equation?Mathematical expressions with two algebraic symbols on either side of the equal (=) sign are called equations.
This relationship is illustrated by the left and right expressions being equal to one another. The left-hand side equals the right-hand side is a basic, straightforward equation.
As per the given information in the question,
Amount spent by Isaiah on gift = $19.60
Let the total amount spent at the store be x.
The amount that he spent on the gift was 5/7 of the total amount that he spent at the store.
So, the equation according to the statement will be,
5/7 of x = $19.60
x = (19.60 × 7)/5
x = 137.2/5
x = $27.44
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Isaiah spent 5/7 of the total amount at the store on a gift. To find the total amount spent, represented by x, we use the equation (5/7) * x = $19.60 and solve for x by multiplying $19.60 by 7/5, which yields $27.44.
Explanation:The question involves finding the total amount spent by Isaiah at the store. Since it is given that $19.60 is 5/7 of the total amount, we can set up the following equation to represent this information: (5/7) * x = $19.60. To find x, we need to perform the inverse operation, which is to divide $19.60 by 5/7, or equivalently to multiply $19.60 by the reciprocal of 5/7, which is 7/5.
Here is the step-by-step solution:
Write down the equation that represents the relationship between the part of the total amount spent on the gift and the whole amount spent: (5/7) * x = $19.60.Multiply both sides of the equation by the reciprocal of 5/7 to solve for x: x = $19.60 * (7/5).Calculate the total amount spent by Isaiah: x = 7 * $19.60 / 5.Complete the computation: x = 7 * 3.92, which gives us x = $27.44.If the range of the function is F(X)=x/4 {28, 30, 32, 34, 36}, what is its domain?
28 x 4 = 112
30 x 4 = 120
32 x 4 = 128
34 x 4 = 136
36 x 4 = 144
domain = {112,120,128,136,144}
Please help me with this question please !!!!!!!!!!!!
FIND X. THE UNITS ARE IN FEET....pLEASE HELP ME ASAP!!!!
f(x) = 2x2 + 1 and g(x) = x2 – 7, find (f – g)(x).
EASY 20 POINTS
Palmers average running speed is 3 kilometers per hour faster than his walking speed. If Palmer can run around a 40-kilometer course in 4 hours, how many hours would it take for Palmer to walk the same course?
The following set of coordinates represents which figure? (7, 10), (4, 7), (6, 5), (9, 8) Parallelogram Rectangle Rhombus Square
Answer:
The figure is a rectangle
Step-by-step explanation:
* Lets explain how to solve the problem
- To prove the following set of coordinates represents which figure
lets find the distance between each two points and the slopes of
the lines joining these points
- The rule of the distance between two point is
[tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/tex]
- The rule of the slope is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
- Remember:
* Parallel lines have same slopes
* The product of the slopes of the perpendicular lines is -1
# points (7 , 10) and (4 , 7)
∵ [tex]d1=\sqrt{(4-7)^{2}+(7-10)^{2}}=\sqrt{18}[/tex]
∵ [tex]m1=\frac{7-10}{4-7}=\frac{-3}{-3}=1[/tex]
# points (4 , 7) and (6 , 5)
∵ [tex]d2=\sqrt{(6-4)^{2}+(5-7)^{2}}=\sqrt{8}[/tex]
∵ [tex]m2=\frac{5-7}{6-4}=\frac{-2}{2}=-1[/tex]
# points (6 , 5) and (9 , 8)
∵ [tex]d3=\sqrt{(9-6)^{2}+(8-5)^{2}}=\sqrt{18}[/tex]
∵ [tex]m3=\frac{8-5}{9-6}=\frac{3}{3}=1[/tex]
# points (9 , 8) and (7 , 10)
∵ [tex]d4=\sqrt{(7-9)^{2}+(10-8)^{2}}=\sqrt{8}[/tex]
∵ [tex]m4=\frac{10-8}{7-9}=\frac{2}{-2}=-1[/tex]
∵ d1 = d3 = √18 and d2 = d4 = √8
∴ Each two opposite sides are equal
∵ m1 = m3 = 1 and m2 = m4 = -1
∴ Each two opposite sides are parallel
∵ m1 × m2 = 1 × -1 = -1
∵ m2 × m3 = 1 × -1 = -1
∵ m3 × m4 = 1 × -1 = -1
∵ m4 × m1 = 1 × -1 = -1
∴ Each two adjacent sides are perpendicular
- The set of coordinates represents a figure has these properties:
1. Each two opposite sides are equal
2. Each two opposite sides are parallel
3. Each two adjacent sides are perpendicular
∴ The figure is a rectangle
1. Compare the strengths and weaknesses of the horizontal and vertical methods for adding and subtracting polynomials. Include common errors to watch out for when using each of these methods.
2. Explain why you cannot use algebra tiles to model the multiplication of a linear polynomial by a quadratic polynomial.
As an added challenge, develop a model similar to algebra tiles that will allow you to show this multiplication. Describe an example of your model for the product (x + 1)(x2 + 2x + 2).
3. Imagine that you are teaching a new student how to multiply polynomials. Explain how multiplying polynomials is similar to multiplying integers. Then describe the key differences between the two.
4. If you multiply a binomial by a binomial, how many terms are in the product (before combining like terms)? What about multiplying a monomial by a trinomial? Two trinomials?
Write a statement about how many terms you will get when you multiply a polynomial with m terms by a polynomial with n terms. Give an explanation to support your statement.
Answer:
1.To add and subtract polynomials, the horizontal technique of deleting parenthesis, collecting like terms, and simplifying is the simplest. When there are negative terms, it gets more difficult since one must ensure that the term remains negative when gathering comparable terms. The vertical approach of building up a box and adding vertically takes longer to set up, but once completed, there is a clear depiction of where all of the similar terms are.
2.Because the product of a linear factor and a quadratic factor is a cubic product, algebra tiles cannot be used to simulate the multiplication of a linear polynomial by a quadratic polynomial.
3.Distribute the first polynomial's terms to the second polynomial's terms. When multiplying two terms together, remember to multiply the coefficients (numbers) and add the exponents. However, with Integers, you multiply two integers with opposite signs.
4.Before combining like terms, there will be four terms. When a monomial is multiplied by a trinomial, the result is six. There will be nine terms in two trinomials.
Step-by-step explanation: