The smallest positive integer a that satisfies the conditions of the problem is a = 2.
The Intermediate Value Theorem states that if a function f(x) is continuous on the interval [a, b] and takes on values f(a) and f(b) at the endpoints a and b, respectively, then for any value c between f(a) and f(b), there exists at least one value x in the interval [a, b] such that f(x) = c.
In this problem, we are asked to find the smallest positive integer a such that the Intermediate Value Theorem guarantees a zero exists between 0 and a for the polynomial f(x) = 3x³ - 4x² - 3x - 1.
To find this value of a, we need to evaluate f(0) and f(a) and look for a value of a such that f(0) and f(a) have opposite signs. This will guarantee the existence of a root of f(x) between 0 and a.
Evaluating f(0), we get:
f(0) = 3(0)³ - 4(0)² - 3(0) - 1 = -1
To find f(a), we need to solve for a using the equation f(a) = 0. We can use a numerical method, such as the Newton-Raphson method or the bisection method, to find the root of f(x) = 0. Using a calculator or a computer program, we find that f(a) = 0 for a ≈ 1.353.
Since f(0) = -1 and f(a) = 0 have opposite signs, the Intermediate Value Theorem guarantees the existence of at least one zero of f(x) between 0 and a = 1.353. Therefore, the smallest positive integer a that satisfies the conditions of the problem is a = 2.
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12. Select all the pairs that are equivalent.
A 5.80; 5+0.08
B (2x1)+(3x):2.3
6.050; six and five thousandths
D eight and seventy-one hundredths; (8X1) + (7 x 100) + (1 x 1,000)
E nine and nine tenths; 9.9
Results:
The equivalent pairs are:
A. 5.80; 5+0.08,
C. 6.050; six and five thousandths, and
E. nine and nine-tenths; 9.9
How do we determine the equivalent expressions?A. 5.80; 5+0.08
The two are equivalent expressions because they both represent the same decimal number. 5.80 can be interpreted as 5 units and 0.80 units, which is the same as 5 units and 0.08 units. So 5.80 is equal to 5+0.08.
C. 6.050; six and five thousandths
The two expressions describe the same decimal number. 6.050 means 6 whole units, 5 hundredths, and 0.005 thousandths, while "six and five thousandths" means the same thing - 6 whole units, 5 hundredths, and 0.005 thousandths.
E. nine and nine tenths; 9.9
They are also equivalent. Nine and nine-tenths mean 9 whole units and 9 tenths, while 9.9 means the same thing - 9 whole units and 9 tenths.
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inverse of f(x)=(x+3)^3
Answer:
( x - 3 ) 3 .
Step-by-step explanation:
Since f−1(f(x))=x f - 1 ( f ( x ) ) = x and f(f−1(x))=x f ( f - 1 ( x ) ) = x , then f−1(x)=3√x+3 f - 1 ( x ) = x 3 + 3 is the inverse of f(x)=(x−3)3 f ( x ) = ( x - 3 ) 3 .
Graph the following features: Slope = − 2 −2 Y-intercept = 3 3
Answer: graphed and attached
Step-by-step explanation:
to graph a y intercept and slope, first find the y intercept on the graph and place your point. here the y intercept is 3, so (0, 3) because x=0 at a y intercept.
next use slope to find your next point. slope of -2 is the same as -2/1 or down -2 and right +1. so from your y intercept of 3, go down -2 and right 1 and place your point (1, 1). do this again for a 3rd point at (2, -1).
graph is attached.
For each equation, decide whether the line it represents is parallel to p, perpendicular to p, or neither.
y=-3x+10
type your answer.....
y-5=-(2+2) type your answer...
type your answer...
-3x+y=1 type your answer.....
The lines are neither parallel nor perpendicular to line p
Deciding whether the line are parallel or notFrom the question, we have the following parameters that can be used in our computation:
y = -3x + 10
As a general rule
Parallel lines have equal slopesPerpendicular lines have their slopes to be opposite reciprocalsUsing the above as a guide, we have the following:
y - 5 = -(2+2) is neither parallel nor perpendicular to y = -3x + 10-3x + y = 1 is neither parallel nor perpendicular to y = -3x + 10Read more about linear relations at
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The volume of a cylinder is 180π cubic meters. The height of the cylinder is 11.25 meters. What is the radius of the cylinder?
Answer :
Radius = 4 meters.Step-by-step explanation :
It's given that :
The volume of a cylinder is 180π cubic meters. The height of the cylinder is 11.25 meters.
Formula of volume of cylinder = πr²h
where,
r denotes radius of the cylinderh denotes height of the cylinder.By substituting the values of values of height and volume in the above formula we'll get the required value of the radius.
[tex]:\implies [/tex] 180π = πr² 11.25
[tex]:\implies [/tex] 180π = r² × 11.25 π
[tex]:\implies [/tex] 180π/11.25 π = r²
[tex]:\implies [/tex] 16 = r²
[tex]:\implies [/tex] r² = √16
[tex]:\implies [/tex] r = 4
Hence, Radius of the cylinder is 4 meters.
―――――――――――――――――――――――Find the slope of a line perpendicular to the line whose equation is 6x + 5y = -30.
Fully simplify your answer.
Answer:
[tex]6x + 5y = - 30[/tex]
[tex]5y = - 6x - 30[/tex]
[tex]y = - \frac{6}{5} x - 6[/tex]
Since the slope of this line is -6/5, the slope of a perpendicular line is 5/6. The slopes of perpendicular lines are negative reciprocals.
Ryan earns 52 dollars per week working part-time at a book store. He makes one dollar more for each book that he sells. The amount, A (in dollars), that Ryan earns in a week if he sells b books is given by the following function.
Answer:
a=52+b
Step-by-step explanation:
he earns $52 without selling books.
since he only earns 1$ per book, there is no need to put it as the coefficient of b since it won't change
therefore,
$52+b=a or a=$52+b
Find the length of the indicated line segment of VX and SU
Questions 17-18
The length of the indicated segments are determined as:
17. VX = 6 18. SU = 15
How to Find the Length of the Indicated Line Segment?Based on the side-splitter theorem, the sides that are intercepted are divided proportionally. Using this theorem, we will find the length of the indicated segments below:
17. 8/20 = VX/15
Cross multiply:
VX * 20 = 8 * 15
VX * 20 = 120
VX = 120/20
VX = 6
18. SU/10 = 12/8
Cross multiply:
SU = 120/8
SU = 15
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Need help finding the area..
The area of the region in this problem is given as follows:
A = 1 square unit.
How to obtain the area of the region?The curve represented in this problem is given as follows:
y = sin(x).
The bounds of the region are given as follows:
x = 0.x = π/2.Hence the area of the region is represented by the definite integral presented as follows:
[tex]\int_{0}^{\frac{\pi}{2}} \sin{x} dx[/tex]
The integral of the sine is the negative cosine, hence the area is:
A = -cos(x), from x = 0 to x = π/2.
Applying the Fundamental Theorem of Calculus, we have that the area is given as follows:
A = -cos(π/2) + cos(0)
A = -0 + 1
A = 1 square unit.
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Two more then a product of 11 and a number
Step-by-step explanation:
( 11 * x ) + 2 where 'x' is a number
if it takes 7 plumbers 15 days to install a gas risers how many plumbers would be needed to install the same riser in 10 days, please such work!!
To install the gas riser in 10 days, a total number of 11 plumbers would be needed.
We can use the formula:
(number of workers) × (time) = (constant amount of work)
Let's let x be the number of plumbers needed to install the gas riser in 10 days. We can set up the equation:
(7 workers) × (15 days) = (x workers) × (10 days)
Simplifying the left side, we get:
105 worker-days = (x workers) × (10 days)
Dividing both sides by 10 days, we get:
10.5 workers = x
Since we can't have a fraction of a worker, we need to round up to the nearest whole number.
Therefore, we would need 11 plumbers to install the gas riser in 10 days.
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A contractor can by two types of wood. she got 45 expensive wood and 60 cheap wood for the same total price. if the 45 expensive wood cost $3.00 more than the 60 cheap wood, then which equation below shows the correct relationship between the 2 types of wood that she purchased. Also (x =price of the 60 cheap wood)
A.45(x+3)=60x
B.45(x-3)=60x
C.60(x-3)=45x
D.60(x+3)=45x
The correct answer is,
C. 60(x - 3) = 45x.
This equation is based on the information that the 45 expensive wood costs $3.00 more than the 60 cheap wood. In other words, if the price of the 60 cheap wood is x, the price of the 45 expensive wood is x + 3.
Since, Algebra involves working with numbers, equations, and variables to solve real-world problems and understand abstract relationships.
Now, To solve this equation, we must use algebra and solve for x.
First, we multiply both sides of the equation by 4/5. This gives us:
48x + 36 = 45x
-36 = 3x
Finally, we divide left side of the equation by 3, giving us the final solution:
x = -12
Therefore, the price of the 60 cheap wood is $12, and the price of the 45 expensive wood is $15.
that is, 60 (x - 3) = 45x
This equation shows the correct relationship between the two types of wood that the contractor purchased.
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The difference in length of a spring on a pogo stick from its non-compressed length when a teenager is jumping on it after θ seconds can be described by the function f of theta equals 2 times sine theta plus radical 2 period
Part A: Determine all values where the pogo stick's spring will be equal to its non-compressed length. (5 points)
Part B: If the angle was doubled, that is θ became 2θ, what are the solutions in the interval [0, 2π)? How do these compare to the original function? (5 points)
Part C: A toddler is jumping on another pogo stick whose length of its spring can be represented by the function g of theta equals 1 minus cosine squared theta plus radical 2 period At what times are the springs from the original pogo stick and the toddler's pogo stick lengths equal? (5 points)
The values of θ for different conditions were calculated with the help of trigonometric functions.
Given that;
The difference in length of a spring on a pogo stick from its non-compressed length when a teenager is jumping on it after θ seconds can be described by the function,
f (θ) = 2 sin θ + √ 2
Hence, We can simplify as;
A) Function is,
f (θ) = 2 sin θ + √ 2
f (θ) = 2 sin θ + √ 2 = 0
2 sin θ + √ 2 = 0
2 sin θ = - √ 2
sin θ = - 1/√2
θ = nπ ± π/4
Where, n = 1, 3, 5, 7, ...
So, At θ = nπ ± π/4 pogo stick's spring will be equal to its non compressed length.
2) If the angle was doubled, the function will look like this,
f (θ) = 2 sin 2θ + √ 2
f (θ) = 2 sin 2θ + √ 2 = 0
2 sin 2θ + √ 2 = 0
2 sin 2θ = - √ 2
sin 2θ = - 1/√2
θ = nπ/2 + π/8
Where, n = 1, 3, 5, 7, ...
n = 1;
θ = π/2 + π/8
θ = 5π/8
n = 2;
θ = π + π//8
θ = 9π/8
Hence, Solutions are, 5π/8, 9π/8, ...
3) g (θ) = 1 - cos²θ + √2
g (θ) = sin²θ + √2
sin²θ + √2 = 2 sinθ + √2
sin²θ - 2 sinθ= 0
sinθ (sinθ - 2) = 0
sinθ = 0
θ = nπ
Where, n = 1, 3, 5, 7, ..
So, θ = nπ , the lengths of springs from the original pogo stick and the toddler's pogo stick are equal.
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given a function,how can the range be determined
To determine the range you need to evaluate and find a maximum/minimum if you can.
How to determine the range?For a function f(x) = y, we define the range as the set of the possible outputs of the function.
So, to find the range of any function, we need to evaluate it and try to find the minimum and maximum values of the function. To do so it is useful to understand the behavior of different types of functions.
For example linear ones have a range of all the real numbers.
Quadratic ones have a range that depends on its vertex, and so on.
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Bao walked 15 miles in 6 hours. What was her walking rate in miles per hour?
Group of answer choices
2.5 miles per hour
2 miles per hour
1 mile per hour
1.5 miles per hour
The solution is, 2.5 miles per hour was her walking rate in miles per hour.
We know that,
Speed is measured as distance moved over time.
The formula for speed is speed = distance ÷ time.
To work out what the units are for speed, you need to know the units for distance and time.
In this example, distance is in metres (m) and time is in seconds (s), so the units will be in metres per second (m/s).
Speed = Distance/ Time.
Here, given that,
Bao walked 15 miles in 6 hours.
so, we get,
in 6 hours = 15 miles
so, in 1 hour = 15 / 6 miles
=2.5 miles
Hence, The solution is, 2.5 miles per hour was her walking rate in miles per hour.
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1/2x+1/3y=7
1/4x+2/3y=6
4
What is the solution of the system shown?
6,12
1/6,14
10 2/3, 5
The solution to the system of linear equation is:
x = 10 2/3, y = 5
The correct option is the last option 10 2/3, 5
Solving system of linear equationsFrom the question, we are to solve the given system of linear equation.
The given system of linear equations is
1/2x + 1/3y = 7
1/4x + 2/3y = 6
Multiply the first equation by 2
1/2x + 1/3y = 7 × 2
x + 2/3y = 14
Subtract the second equation from the resulting equation
x + 2/3y = 14
- 1/4x + 2/3y = 6
--------------------------
3/4x = 8
Multiply both sides of the equation by 4/3
4/3 × 3/4x = 8 × 4/3
x = 32/3
x = 10 2/3
Substitute the value of x into
x + 2/3y = 14
To determine y
(10 2/3) + 2/3y = 14
32/3 + 2/3y = 14
Multiply through by 3
32 + 2y = 42
Subtract 32 from both sides
32 - 32 + 2y = 42 - 32
2y = 10
Divide both sides by 2
2y / 2 = 10 / 2
y = 5
Hence, the solution is:
x = 10 2/3, y = 5
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Joshua and Amahle are selling boxes of fruit for a Habitat for Humanity fundraiser. Customers can buy small
boxes of fruit and large boxes of fruit. Joshua sold 15 small boxes of fruit and 15 large boxes of fruit for a
total of $390.00. Amahle sold 20 small boxes of fruit and 25 large boxes of fruit for a total of $592.50. Find the cost of one small box of fruit and the cost of one large box of fruit.
Cost of small box: $
Cost of large box: $
The cost of one small box of fruit and the cost of one large box of fruit is $11.5 and $14.5 respectively.
What is the cost of small and large box?Let
cost of one small box of fruit = x
cost of one large box of fruit = y
15x + 15y = 390
20x + 25y = 592.50
From (1)
x + y = 26
x = 26 - y
Substitute x = 26 - y into (2)
20x + 25y = 592.50
20(26 - y) + 25y = 592.50
520 - 20y + 25y = 592.50
- 20y + 25y = 592.50 - 520
5y = 72.50
divide both sides by 5
y = 72.50/5
y = 14.5
Substitute y = 14.5 into
15x + 15y = 390
15x + 15(14.5) = 390
15x + 217.5 = 390
15x = 390 - 217.5
15x = 172.5
divide both sides by 15
x = 172.5/15
x = 11.5
Therefore, small box cost $11.5 and large box cost $14.5
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A(t)=600e^-0.05t
Find the half life of the material to the nearest 10th of a day
The half life of the decaying material is found to be 13.9 days to the nearest 10th of a day.
Using the radioactivity formula given to find out the concertation and the half life of the material. In this case, we are given the function A(t) = 600e^(-0.05t), which represents the quantity of the material at time t.
Let A(0) be the initial amount of the material, then we have,
[tex]A(0) = 600e^{-0.05(0)} = 600[/tex]
To find the half-life, we need to solve for t when A(t) = A(0)/2:
A(t) = A(0)/2
[tex]600(e^{-0.05t}) =\frac{600}{2}[/tex]
Taking natural logarithm,
[tex]ln(e^{-0.05t}) = ln(\frac{1}{2})[/tex]
-0.05t = ln(1/2)
t = -ln(1/2)/0.05
t ≈ 13.9
Hence, the half life is found to be about 13.9 to the nearest tenth of the day.
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The function d = 45t gives the distance d in miles a car travels in t hours. Graph this function
A graph of this function d = 45t is shown in the image below.
What is a proportional relationship?In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:
d = kt
Where:
d represents the distance in miles.t represents the time in hours.k is the constant of proportionality.Next, we would determine the constant of proportionality (k) by using various data points as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 45/1 = 90/45
Constant of proportionality, k = 45.
Therefore, the required linear equation or function is given by;
d = kt
d = 45t
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What is the length of the unknown leg of the triangle?5 ft 8 ft
Let z = 3( cos pi/2 + i sin pi/2). Find the exact value of z^7 where 0 (less than equal to) theta (less than equal to) 2pi.
From the given value of z = 3(cos pi/2 + isin pi/2), the exact value of z⁷ is -2187i.
Given z = 3(cos pi/2 + isin pi/2)
which can be represented as 3sin(pi/2) in polar form.
Now, by applying De-Moivre's theorem,
z⁷ = (3cis(pi/2))⁷
= 3⁷cis(7pi/2)
= 3⁷cis(3pi/2) [7pi/2 --- 3pi/2 ]
To write it in complex form,
z⁷ = 3⁷(cos(3pi/2) + isin(3pi/2))
z⁷ = 3⁷(0-i) [cos(3pi/2) = 0 and sin(3pi/2) = -1]
z⁷ = 3⁷(-i)
z⁷ = -2187i
From the above explanation, we can conclude that the value of z⁷ is -2187i.
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solve for x ( needed quickly)
Wooden blocks in the shape of
cubes are painted with different
colors. How many square
centimeters will be painted?
4 cm
The total area of all six sides that will be painted on the wooden cube is 96 square centimeters.
First, we need to understand that each cube has six faces or sides, which are all squares of the same size. When we paint the cube, we will be painting all six sides of it. Therefore, we need to find the total area of all six sides that will be painted.
To calculate the area of each side, we need to know the length of the side. You mentioned that the wooden blocks are cubes, so each side will have the same length. Let's say the length of each side is 4 cm.
Now, we can calculate the area of one side of the cube by squaring the length of the side. In this case, the area of one side will be:
Area of one side = (length of side)²
= (4 cm)²
= 16 cm²
Since there are six sides to each cube, we can find the total area of all six sides by multiplying the area of one side by six:
Total painted area = 6 x Area of one side
= 6 x 16 cm²
= 96 cm²
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is abc ~ Def explain
The two triangles are congruent since they have equal angles and equal sides.
What is congruent triangles?Congruent triangles are two triangles that have exactly the same size and shape.
In other words we can say that they have the same angles and the same side lengths.
So when two triangles are congruent, all of their corresponding parts including the angles and sides are equal.
For the given triangles
The missing value of angle C = 180 - (18 + 110) = 52⁰
The missing value of angle F = 180 - (18 + 52) = 110⁰
So it is obvious that the two triangles are congruent since they have equal angles and equal sides.
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What is the product of the rational expressions below?
6-X 6+** 2-X Zt
The value of the product of the rational expressions is,
⇒ (4x - 32) (x² - 36)
We have to given that;
The rational expressions is,
⇒ 4x/(x+6) times (x-8) / (x-6)
Now, We can multiply as;
⇒ 4x/(x+6) times (x-8) / (x-6)
⇒ 4x/(x+6) × (x-8) / (x-6)
⇒ 4 (x - 8) / (x + 6) (x - 6)
⇒ (4x -32) / (x² - 6²)
⇒ (4x - 32) (x² - 36)
Thus, The value of the product of the rational expressions is,
⇒ (4x - 32) (x² - 36)
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Suppose there are four shelves of books in a library with 12 books, 18 books, 25 books, and 30 books respectively. In how many ways can a person choose three books, one from each shelf? Explain the rule used to solve the problem.
Answer:
Step-by-step explanation:
To solve this problem, we can use the multiplication principle of counting, which states that if there are k choices for one task and m choices for a second task, then there are k x m choices for both tasks combined.
In this case, we can apply this principle to each shelf of books, as there are different numbers of books on each shelf. For the first shelf, there are 12 choices, for the second shelf there are 18 choices, for the third shelf there are 25 choices, and for the fourth shelf there are 30 choices.
To choose one book from each shelf, we need to multiply the number of choices for each shelf together, which gives us:
12 x 18 x 25 x 30 = 16,200,000
Therefore, there are 16,200,000 ways to choose three books, one from each shelf, in the library.
The total number of ways of choosing three books , one from each shelf is given by A = 162,000 ways
What are Combinations?The number of ways of selecting r objects from n unlike objects is given by combinations
ⁿCₓ = n! / ( ( n - x )! x! )
where
n = total number of objects
x = number of choosing objects from the set
Given data ,
There are 12 ways to choose a book from the first shelf, 18 ways to choose a book from the second shelf, 25 ways to choose a book from the third shelf, and 30 ways to choose a book from the fourth shelf
So , the total number of ways = 12 x 18 x 25 x 30
A = 162,000 ways
Hence , the combination is solved and there are 162,000 ways
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Rosanna sells m small bags of marbles and p large bags of marbles.
Each small bag contains 15 marbles.
Each large bag contains 40 marbles.
The total number of marbles that Rosanna sells is T
Write down a formula for T in terms of m and p
The formula for T in terms of m and p is: [tex]T = 15m + 40p[/tex]
How do we determine the T in terms of m & p?We are given that number of marbles in each small bag is 15, and Rosanna sells m small bags, so the total number of marbles in the small bags is: [tex]15m[/tex]
Also given that number of marbles in each large bag is 40, and Rosanna sells p large bags, so the total number of marbles in the large bags is:[tex]40p[/tex]
So, total number of marbles that Rosanna sells is the sum of the marbles in the small bags and in the large bags will be addition of both values which will give us: [tex]T = 15m + 40p[/tex].
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matet please
-6+28÷(-4)
Answer:
-13
Step-by-step explanation:
-6 + 28 ÷ (-4)
= -6 - 7
= -13
So, the answer is -13
-5|x-8|+2=2 pls help me pls
Answer:
x = 8
Step-by-step explanation:
[tex]5|x-8|+2=2 \\ −5∣x−8∣=0 \\ ∣x−8∣=0 \\ x = 8[/tex]
First, you need to cancel out the 2 by subtracting 2 from both sides. Then, we divide both sides by -5. And finally, we remove the absolute value.
when will 65x +20= y and 730+35x=y meet on the graph?
To find when the two equations 65x + 20 = y and 730 + 35x = y will meet on the graph, we need to solve for the value of x that satisfies both equations.
We can begin by setting the right-hand sides of the two equations equal to each other:
65x + 20 = 730 + 35x
Subtracting 35x from both sides, we get:
30x + 20 = 730
Subtracting 20 from both sides, we get:
30x = 710
Dividing both sides by 30, we get:
x = 23.67
Now that we have found the value of x that satisfies both equations, we can substitute it into either equation to find the corresponding value of y. Let's use the first equation:
y = 65x + 20
y = 65(23.67) + 20
y = 1539.55
Therefore, the two equations will meet on the graph at the point (23.67, 1539.55).