By solving an expression, we can find that the number of packs of paper that each classroom should get is 10/3.
Define Expression.Expressions are mathematical statements that comprise either numbers, variables, or both and at least two terms associated by an operator. Addition, subtraction, multiplication, and division are examples of mathematical operations. An example is the expression x + y, which combines the terms x and y with an addition operator. In mathematics, there are two different types of expressions: algebraic expressions, which also include variables, and numerical expressions, which solely comprise numbers.
Given,
Number of classrooms= 3
Number of packs of paper= 10
Now let the number of packs of paper per class be x.
x= Number of packs of paper/Number of classrooms
= 10/3.
Therefore, each classroom will get 10/3 packs of paper.
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An ecologist wishes to mark off a circular sampling region having radius 13 m. However, the radius of the resulting region is actually a random variable R with pdf given below. s(r) = { 1-(13-r)?] 12
The ecologist needs to carefully choose the size of the sampling region based on the pdf of R.
Sampling is the process of selecting a representative subset of individuals or objects from a larger population to study or analyze.
To ensure that their data is representative of the entire ecosystem, it is important to choose a sampling region that is large enough and well-defined.
In this scenario, an ecologist wishes to mark off a circular sampling region with a fixed radius of 13 meters. However, due to natural variation, the actual radius of the sampling region is a random variable, denoted by R. The probability density function (pdf) of R is given as
=> s(r) = { 1-(13-r)] ¹²,
which means that the probability of a particular value of R is proportional to the area of a circle with that radius.
Now, when the ecologist takes a sample, it is important to consider the effects of this random variation in the sampling region. If the radius of the sampling region is too small, it may not capture the full range of variability in the ecosystem.
On the other hand, if the radius is too large, it may include areas that are not representative of the ecosystem as a whole.
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Do any ingredients make up an equal amount of the smoothie? Construct a math argument to explain why or why not
Without knowing the specific ingredients and proportions used in a smoothie, it is impossible to definitively say whether any ingredients make up an equal amount. However, it is likely that the proportions of ingredients in a smoothie vary based on the desired taste, texture, and nutritional content.
For example, a smoothie may have more fruit than liquid to create a thicker consistency, or more liquid than fruit to create a thinner consistency. Additionally, the amount of sweeteners, such as sugar or honey, may vary based on personal preference and the natural sweetness of the fruit used.
In conclusion, without more specific information about the ingredients and proportions used in a particular smoothie, it is not possible to determine whether any ingredients make up an equal amount. The proportions of ingredients in a smoothie are likely to vary based on various factors and personal preference.
3(6-4)=-3
How do I show work for this
The expression is not true.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ 3 (6 - 4) = - 3
Now, We can check as;
⇒ 3 (6 - 4) = - 3
LHS;
⇒ 3 (6 - 4)
⇒ 3 × 2
⇒ 6
RHS;
⇒ - 3
Hence, The expression is not true.
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What is the center and radius of the circle with equation (x + 5)2 + (y – 3)2 = 4?
Choose one
(5, –3), r = 16
(5, –3), r = 4
(–5, 3), r = 16
(–5, 3), r = 4
none of these
The center and radius of the circle in the equation is given by (-5, 3) and 4 respectively.
What is circle?A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.
Given that, an equation of a circle, (x+5)²+(y-2)² = 4, we need to identify the center and radius of the circle,
The standard form for the equation of a circle is (x−h)²+(y−k)² = r². The center is (h, k) and the radius measures r units.
Comparing with the standard form, we get,
(h, k) = (-5, 2) and r² = 4
Therefore, the center is (-5, 2) and the radius is r = 4
Hence, the center and radius of the circle in the equation is given by (-5, 3) and 4 respectively.
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Tara used a credit card to pay for a jacket that costs
$150. She did not purchase any other items using the
credit card. The credit card company charges 25%
interest each month on the remaining balance. Tara
pays $40 per month after the monthly interest has
been applies. How long will it take Tara to pay off the
balance? How much does she pay in all?
It takes Tara 5 months to pay off the balance.
What is the rate of interest?An interest rate tells you how high the cost of borrowing is, or high the rewards are for saving. So, if you're a borrower, the interest rate is the amount you are charged for borrowing money, shown as a percentage of the total amount of the loan.
Given that, Tara used a credit card to pay for a jacket that costs $150.
The credit card company charges 25% interest each month on the remaining balance.
Now, 25% of 150
25/100 ×150
0.25×150= 37.5
So, total money is 150+37.5
= 187.5
Number of months = 187.5/40
= 4.6875
≈ 5
Therefore, it takes Tara 5 months to pay off the balance.
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eleven spoons minus s equals 9 spoons
Answer:
s=2
Step-by-step explanation:
Please helpppppppppppppppp
An office manager believes that the percentage of employees arriving late is even greater than the previously claimed 7%. She conducts a hypothesis test on a random 200 employee arrivals and finds 23 punching in late. Is this strong evidence against the 0.07 claim?
answer choices
Yes, p-value is 0.0062
Yes, p-value is 2.5
No, p-value is only 0.0062
No, p-value is over 0.10
There is insufficient information to reach a conclusion.
We do not have strong evidence against the 0.07 claim so, the correct answer is: No, the p-value is over 0.10.
Let's calculate the p-value and compare it to the level of significance (alpha) to determine if there is strong evidence against the 0.07 claim.
Let p be the true proportion of employees arriving late. The null hypothesis is that p = 0.07 and the alternative hypothesis is that p > 0.07.
The sample proportion of employees arriving late is x/n = 23/200 = 0.115. Under the null hypothesis, the sampling distribution of the sample proportion follows a normal distribution with a mean of 0.07 and standard deviation [tex]\sqrt{((0.07 * 0.93) / 200)}[/tex] = 0.0319.
The z-score corresponding to the sample proportion is (0.115 - 0.07) / 0.0319 = 1.41. The p-value is the probability of observing a z-score greater than or equal to 1.41 under the null hypothesis, which can be calculated as P(Z >= 1.41) = 0.078.
Since the p-value is greater than the level of significance alpha (which is typically 0.05 or 0.01), we fail to reject the null hypothesis that the true proportion of employees arriving late is 0.07.
Therefore, we do not have strong evidence against the 0.07 claim so, the correct answer is: No, the p-value is over 0.10.
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Use the drop-down menus to complete each equation so the statement about its solution is true.
The illustration of the types of solutions of systems of equations is explained below.
What is an equation?An equation is written in the form of variables and constants separated by the operation of multiplication and division,
An equation states that terms in different forms on both sides of the equality sign are equal.
Multiplication and division do not separate the terms of an equation.
Given, 5 - 4 + 7x + 1 = ax + b.
7x + 2 = ax = b.
Now, If we put b ≠ 2 and x = 7 we have,
7x + 2 = 7x + b.
2 = b, but b ≠ 2 so no solution.
5 - 4 + 7x + 1 = ax + b
7x + 2 = ax + b.
To have one solution we can assume any x value except 7 and any b value except 2,
7x + 2 = 6x + 3.
x = - 1.
5 - 4 + 7x + 1 = ax + b
To have infinitely many solutions both sides must be some constant multiple of each other.
7x + 2 = k(ax + b).
7x + 2 = 2(7x + 2).
7x + 2 = 14x + 4.
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You need to rent a moving truck for a day. You have identified two companies that rent trucks. Company A charges $45 per day plus $0.10 per mile. Company B charges $25 per day plus $0.30 per mile. For how many miles will the cost of renting be the same?
Answer:
the cost of renting a truck from both companies will be the same for 100 miles.
Step-by-step explanation:
Let's assume that the cost of renting a truck from Company A for x miles is C_A(x) and the cost of renting a truck from Company B for x miles is C_B(x).
From the given information, we know that:
C_A(x) = 45 + 0.10x
C_B(x) = 25 + 0.30x
We need to find the value of x for which C_A(x) = C_B(x). So we can set the two equations equal to each other and solve for x:
45 + 0.10x = 25 + 0.30x
20 = 0.20x
x = 100
Therefore, the cost of renting a truck from both companies will be the same for 100 miles.
3 tickets to the museum cost $12.75. At this rate, what is the cost of:
A. 1 ticket?
B. 5 tickets?
Answer:
1 ticket - $4.25
5 tickets - $21.25
Step-by-step explanation:
12.75/3=4.25 which is one ticket then take the price of one ticket and multiply it by 5, 4.25*5=21.25.
find two numbers whose product is 15 and sum is 16
Answer:
Step-by-step explanation:
l
Answer:
15+1
Step-by-step explanation:
Kellan went ice skating for 8 hours this week. If he spent an equal amount of time skating each day for 5 days, how many hours did kellan skate each day?
Answer:
8/5 = 1.6
Step-by-step explanation:
8/5 = 1.6
1) Consider the second-order linear DE (1- tcot t) y - ty y 0 for 0 < t < T- Let Y1 (t)-t and y2 (t) sin t. a) Are y1 and y2 both solutions to this DE? b) Are yi and y2 linearly independent? If so, then find the general solution of the DE. If not, then find constants A and B, not both zero, such that Ay? By 0. 2) Suppose that is a constant and {31 , y2} is a set of solutions to the DE Find the Wronskian W(x) of {31 , y, given that W(0) = 1.
The solution of second order linear equation DE (1- tcot t) y - ty y 0 for 0 < t < T is y2(t) = sin(t). There cannot be exist a constant k such that {x^2, e^x} is a set of solutions to the given differential equation.
To check if y1(t) = t is a solution to the given differential equation, we substitute it into the DE and simplify:
(1 - t cot(t))y - t(dy/dt) = (1 - t cot(t))t - t(1) = t - t^2 cot(t) - t = t(1 - t cot(t))
So y1(t) = t is a solution to the DE.
To check if y2(t) = sin(t) is a solution, we substitute it into the DE and simplify:
(1 - t cot(t))y - t(dy/dt) = (1 - t cot(t))sin(t) - t(cos(t)) = sin(t) - t sin(t) (csc(t)) - t(cos(t)) = sin(t) - t tan(t) sin(t) - t(cos(t))
So y2(t) = sin(t) is also a solution to the DE.
To check if y1(t) and y2(t) are linearly independent, we can check if their Wronskian is nonzero. The Wronskian of two solutions y1(t) and y2(t) of a second-order linear differential equation is given by:
W(y1, y2) = y1(t) (dy2/dt) - y2(t) (dy1/dt)
Substituting y1(t) = t and y2(t) = sin(t), we get:
W(y1, y2) = t cos(t) - sin(t)
Since W(y1, y2) is not identically zero (it is zero only at certain values of t), we can conclude that y1(t) and y2(t) are linearly independent.
The general solution to the given differential equation is given by:
y(t) = c1 t + c2 sin(t)
where c1 and c2 are constants determined by the initial or boundary conditions.
Let W(x) be the Wronskian of the solutions y1(x) and y2(x) of the differential equation given by:
(1 - x^2) y'' - 2x y' + k y = 0
Then, we have:
W(x) = y1(x) y2'(x) - y1'(x) y2(x)
Substituting y1(x) = x^2 and y2(x) = e^x, we get:
W(x) = x^2 e^x - 2x e^x - x^2 e^x = -2x e^x
Since we are given that W(0) = 1, we have:
W(0) = -2(0) e^0 = 1
This gives us a contradiction, so there cannot exist a constant k such that {x^2, e^x} is a set of solutions to the given differential equation.
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Find the radius of a sphere with a volume of 26, 244π cubic inches.
27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
Given that volume of a sphere is 26, 244π cubic inches.
We need to find the radius of sphere.
V=4/3πr³
26, 244π=4/3πr³
26, 244=1.333r³
Divide both sides by 1.333
26,244/1.333=r³
19688=r³
Cube root on both sides
∛19688 = r
27=r
Hence, 27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.
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Determine where f is continuous. If possible, extend f as in example 4.2 to a new function that is continuous on a larger domain.F(x) = {sin x/x if x ≠ 0; 1 if x= 0Continuous everywhere since limx→0 sinx/x=1,and f(0) = 1.
This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. It cannot be extended to a larger domain, as it is already continuous everywhere.
For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667.This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. This means that there are no breaks or discontinuities in the function, and it is continuous everywhere. To demonstrate this, we can look at specific values of x. For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667. This shows that the function is smoothly transitioning from one value to the next, and is therefore continuous everywhere. The function cannot be extended to a larger domain, as it is already continuous everywhere. This means that the function is defined everywhere on its domain, and there is no need to extend it.
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NO LINKS!! URGENT HELP PLEASE!!
For #4-6, find the area of each figure, round your answer to one decimal point if necessary.
Problem 4
Answer: 2400 square inchesExplanation:
Draw a horizontal divider line to split the figure into two rectangles.
The rectangle up top is 30 by 30, so it has area 30*30 = 900 square inches.
The rectangle on the bottom has area 50*30 = 1500 square inches.
total = 900+1500 = 2400 square inches
========================================================
Problem 5
Answer: 12 square inchesExplanation:
Draw a horizontal divider line to split the figure into two rectangles.
A = area of rectangle on top = 2*2 = 4 square inches
B = area of rectangle on bottom = 4*2 = 8 square inches
C = total = A+B = 4+8 = 12 square inches
========================================================
Problem 6
Answer: 2592 square yardsExplanation:
Draw a horizontal divider line to get two rectangles.
A = area of rectangle on top = 36*36 = 1296 square yards
B = area of rectangle on bottom = 72*18 = 1296 square yards
C = total = A+B = 1296+1296 = 2592 square yards.
You have the correct answer. Nice work.
Answer:
4) 2400 in²
5) 12 in²
6) 2592 yd²
Step-by-step explanation:
To calculate the area of each given composite figure, divide the figure into two rectangles and sum the area of the two rectangles.
[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]
Question 4Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 30\cdot 30+50\cdot 30\\&=900+1500\\&=2400\; \sf in^2\end{aligned}[/tex]
Question 5Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=2 \cdot 2+ 4\cdot2 \\&=4+8\\&=12\; \sf in^2\end{aligned}[/tex]
Question 6Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=36 \cdot36 +72 \cdot18 \\&=1296+1296\\&=2592 \; \sf yd^2\end{aligned}[/tex]
Use tape diagram to show Noah saved x dollars and Elena saved 1/10 less than that
What was the first step in solving this equation?
Original equation:
9(2p+ 1) = 6p + 15
Fill in the missing work:
Result of the first step:
>
3(2p + 1) = 2p + 5
Write a few sentences describing the missing work.
The missing step is 9(2p+ 1)/3 = 6p/3 + 15/3.
What is division?
The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
An original equation,
9(2p+ 1) = 6p + 15.
Result of the first step:
3(2p + 1) = 2p + 5
To find the missing step:
Divide the original equation by 3 into both sides,
we get,
9(2p+ 1)/3 = 6p/3 + 15/3.
Simplifying further,
3(2p + 1) = 2p + 5
Therefore, the required expression is 9(2p+ 1)/3 = 6p/3 + 15/3.
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A parachutist whose mass is 100 kg drops from a helicopter hovering 3000 m above the ground and falls under the influence of gravity. Assume that the force due to air resistance is proportional to the velocity of the parachutist, with proportionality constant b3 = 20 N-sec/m when the chute is closed and b4 = 100 N-sec/m when the chute is open. If the chute does not open until 30 sec after the parachutist leaves the helicopter, after how many seconds will he hit the ground? If the chute does not open until 1 min after he leaves the helicopter, after how many seconds will he hit the ground?
If the chute does not open until 1 min after he leaves the helicopter, then the equation to calculate the seconds it will he hit the ground is (1/2)mv² + ∫ F(v) dv = 0
The force due to air resistance is assumed to be proportional to the velocity of the parachutist, and we will use this information to determine the time it takes for the parachutist to reach the ground with and without a chute.
The force of air resistance is proportional to the velocity of the parachutist. This means that the larger the velocity, the larger the force of air resistance.
As the velocity increases, the force of air resistance also increases, which reduces the kinetic energy. When the chute opens, the force of air resistance increases even more, which further reduces the kinetic energy and slows down the parachutist.
To determine the time it takes for the parachutist to hit the ground, we can use the following equation:
PE + KE + work done by air resistance = PE + KE
where PE is the gravitational potential energy, KE is the kinetic energy, and work done by air resistance is the work done by the force of air resistance.
When the chute is closed, we can write:
mgh = (1/2)mv² + ∫ F(v) dv
where m is the mass of the parachutist, g is the acceleration due to gravity, h is the height of the helicopter above the ground, and the integral is taken over the velocity range from 0 to v.
Similarly, when the chute is open, we can write:
mgh = (1/2)mv² + ∫ F(v) dv
where b = b4 and the integral is taken over the velocity range from 0 to v.
To solve for the time it takes for the parachutist to hit the ground, we need to determine the velocity at which he hits the ground. This is the velocity where KE = 0, and we can find it by equating the expressions for KE and work done by air resistance when the parachutist hits the ground:
(1/2)mv² + ∫ F(v) dv = 0
We can solve these equations numerically to find the time it takes for the parachutist to hit the ground with and without the chute.
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Find all other zeros of P(x)=x^3-6x^2+18x-40 , given that 1+3i is a zero.
Answer:
1 - 3i, 4
Step-by-step explanation:
The polynomial has real coefficients, so if it has a complex root, it must have at least two complex roots which are complex conjugates.
Since 1 + 3i is a root, then 1 - 3i is also a root.
Two of the factors of the polynomial are:
x - (1 + 3i) and x - (1 - 3i)
Simplify the factors above:
x - 1 - 3i and x - 1 + 3i
Find their product:
(x - 1 - 3i)(x - 1 + 3i) =
Rewrite them showing we have the product of a sum and a difference.
= [(x - 1) - 3i][(x - 1) + 3i]
Multiply the factors above noticing they are a sum and a difference which follows the pattern (a + b)(a - b) = a² - b².
= (x - 1)² - (3i)²
= x² - 2x + 1 - 9(-1)
= x² - 2x + 10
Now we divide the original polynomial by the product we just found using long division.
x - 4
------------------------------
x² - 2x + 10 | x³ - 6x² + 18x - 40
- x³ - 2x² + 10x
------------------------
-4x² + 8x - 40
- -4x² + 8x - 40
--------------------------
0 + 0 + 0
Now we know that x³ - 6x² + 18x - 40 factors into (x² - 2x + 10)(x - 4).
(x² - 2x + 10)(x - 4) = 0
From x² - 2x + 10, we have roots x = 1 + 3i and x = 1 - 3i.
x - 4 = 0
x = 4
From x - 4, we have root x = 4.
Answer: 1 - 3i, 4
Complete the table:
0
Cos(8)
a. 1
b. -1
?
90
10
C. 2
d.
0
a. 1
b. -1
?
90
10
c. 0
d. -2
A construction crew needs to pave a road that is 204 miles long. The crew paves 7 miles of the road each day. The length, L (in miles) that is left to be paved after d days is given by the following function. L (d) =204-7. If 155miles of road is left to be paved how many days has the crew been paving the road? How many miles of road does the crew have left to pave after 13days?
The number of days required to pave 155 miles of the road will be 7 days. The length of the road paved in 13 days will be 113 miles.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the length, L (in miles) that is left to be paved after d days is given by the following function. L (d) =204-7d.
The number of days will be calculated as:-
L (d) =204-7d
155 = 204 - 7d
7dv = 49
d = 7 days
The length of the road paved in 13 days will be calculated as:-
L (d) =204-7d
L (d) =204-7 x 13
L (d) =113 miles
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Calculate the area of this rectangle.
1/2 ft.
3
ft.
The area of a rectangle with the given length breadth is 1.5 square feet.
What is the area of a rectangle?The area occupied by a rectangle within its boundary is called the area of the rectangle. The formula to find the area of a rectangle is Area = Length × Breadth.
Given that, length of a rectangle= 3 ft and width of a rectangle= 1/2 ft.
Area is defined as the total space taken up by a flat surface or shape of an object.
Here, area of a rectangle =Length × Breadth
= 3×1/2
= 3×0.5
= 1.5 square feet
Therefore, the area of a rectangle is 1.5 square feet.
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A family wants to rent a car to go on vacation. Company A charges $55.50 and 16 cents per mile. Company B charges $50.50 and 9 cents per mile. How much more does Company A charge for x miles than Company B?
Company A will charge $7 more than company B.
What is an expression?Expression in maths is defined as the relation of numbers variables and functions by using mathematical signs like addition, subtraction, multiplication and division.
Given that a family wants to rent a car to go on vacation. Company A charges $55.50 and 16 cents per mile. Company B charges $50.50 and 9 cents per mile.
The expression for the charges by company A is,
C = 16x + 55.50
The expression for the charges by company B,
C = 9x + 55.50
The amount of company A more than company B is calculated as:-
Amount = ( 16 + 55.50) - ( 9 + 55.50 )
Amount = 71.50 - 64.50
Amount = $7
Hence, company A will charge $7 more.
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Cost of headphones: $20.00 Tax: 6% What is the new cost?
Answer:
Step-by-step explanation:
e = 20 * (1+6/100) = 20 * 53 50 = 20 * 1.06 =$ 21.2
Jack tenía 3 bolsas de pelotas de golf con b pelotas en cada bolsa; luego su amigo le dio 6
pelotas de golf más.
¿Cuántas pelotas de golf tiene ahora Jack?
The total number of golf balls with Jack as -
{n} = 3b + 6.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Given is that Jack had 3 bags of golf balls with {b} balls in each bag; then his friend gave him 6 golf balls more.
We can write the total number of golf balls with Jack as -
{n} = 3b + 6
Therefore, the total number of golf balls with Jack as -
{n} = 3b + 6.
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{Question in english -
Jack had 3 bags of golf balls with b balls in each bag; then his friend gave him 6 golf balls more. How many golf balls does Jack have now?}
Carmen planted some corn in the back yard and measures its height regularly. The last time she checked, the stalks were 52.6 inches tall. Now they are the same height. What is the percent of increase in the height of the stalks?
The percent of increase in the height of the stalks is given as follows:
0%.
How to obtain the percent of increase?The percent of increase is obtained applying the proportions in the context of this problem.
A proportion is applied because the percent increase is given by the division of the increase by the initial amount, and multiplied by 100%.
The parameters for this problem are given as follows:
Change of 0, as the plant is of the same height as the initial height.Initial height of 52.6 inches.Hence the percent increase is given as follows:
0/52.6 x 100% = 0%.
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A) In the sequence for which the general term is t_(n)=(n^(2)-1)/(3n^(2)),
find the value of n, if any, for which t_(n)=0.3325.
B) Find S_(n) for the arithmetic series 16+13+10+cdots and determine the value of n for which S_(n)=-6.
S_(n)=. The value of n for which S_(n)=-6 is .
we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4 and the value of n for which [tex]S_n[/tex]= -6 is 6.
How to find value of n and sum of sequence?A) To find the value of n for which [tex]t_n[/tex] = 0.3325, we can use trial and error method.
Starting with n=1, we have:
[tex]t_1 = (1^2 - 1)/(3 * 1^2) = 0[/tex]
For n=2, we have:
[tex]t_2 = (2^2 - 1)/(3 * 2^2) = (3)/(12) = 0.25[/tex]
For n=3, we have:
[tex]t_3 = (3^2 - 1)/(3 * 3^2) = (8)/(27) = 0.296296...[/tex]
Continuing in this way, we find that t_3 < 0.3325 < t_4, where
[tex]t_4 = (4^2 - 1)/(3 * 4^2) = (15)/(48) = 0.3125[/tex]
Thus, we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4, and it can be found more accurately by using numerical methods such as bisection or Newton's method.
B) To find the sum of the arithmetic series 16 + 13 + 10 + ..., we can use the formula for the sum of an arithmetic series:
[tex]S_n = n/2 * (a_1 + a_n),[/tex]
where n is the number of terms in the series, [tex]a_1[/tex] is the first term (16 in this case), and a_n is the nth term.
Let's assume that the nth term is represented by [tex]a_n = 16 - 3 * (n - 1),[/tex] so that a_1 = 16 and a_n = 16 - 3 * (n - 1).
Then, we have:
S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n
We want to find the value of n for which S_n = -6, so we can substitute the value of S_n into the equation and solve for n:
[tex]S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n[/tex]
[tex]-6 = 16n - (3/2)n^2 + (3/2)n[/tex]
Expanding the equation and solving for n, we find that n = 6.
Thus, the value of n for which [tex]S_n[/tex]= -6 is 6.
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