A cylinder has a volume of cubic centimeters and a height of 12 centimeters. What is the radius of the base of the cylinder, in centimeters?"
Answer:
Step-by-step explanation:
A company is designing a new cylindrical water
bottle. The volume of the bottle will be 170 cm³.
The height of the water bottle is 8.1 cm. What is
the radius of the water bottle? Use 3.14 for л.
Height: 8.1 cm
Answer: around 2.6 cm because I rounded to the tenth.
Step-by-step explanation:
r^2=170/8.1×3.14
r^2=170/25.434
r^2≈6.68
Next square root both sides so r^2 becomes r and 6.68 square rooted is about 2.6 cm is the radius.
R≈2.6cm
A coin is flipped, and a standard number cube is rolled. What is the probability for flipping tails and rolling an odd number.
Answer:
1/4
Step-by-step explanation:
The probability of flipping tails is 1/2, since there are two equally likely outcomes when flipping a coin (heads or tails).
The probability of rolling an odd number on a standard number cube is 3/6 or 1/2, since there are three odd numbers (1, 3, and 5) out of six possible outcomes (1, 2, 3, 4, 5, and 6).
To find the probability of both events happening (i.e., flipping tails and rolling an odd number), we multiply the probabilities of each event:
P(tails and odd number) = P(tails) * P(odd number)
P(tails and odd number) = 1/2 * 1/2
P(tails and odd number) = 1/4
Therefore, the probability of flipping tails and rolling an odd number is 1/4 or 0.25.
What percent of his monthly budget do his transportation costs account for?
To calculate the percentage of one's monthly budget that transportation costs account for, we need to know the total amount of money spent on transportation and the total monthly budget.
Let's say, for example, that John spends $500 per month on transportation and his monthly budget is $2,000.
To calculate the percentage, we would divide the amount spent on transportation by the total monthly budget and then multiply by 100 to get the percentage. So, in this case, the calculation would be:
[tex]($500 / $2,000) x 100 = 25%[/tex]
Therefore, John's transportation costs account for 25% of his monthly budget. This is a significant portion of his budget, and if he needs to save money, he may want to consider alternative modes of transportation such as carpooling,
public transportation, or biking. It's always important to keep track of expenses and prioritize spending in order to maintain a healthy financial situation.
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Is Y= 4x^3+6....
A. Linear
B. Nonlinear
C. Both
D. Neither
Answer:
The given equation Y=4x^3+6 is a nonlinear equation because it contains a term with a power of 3, which means that the relationship between Y and x is not linear. In a linear equation, the power of the variable is always 1. Therefore, the answer is B. Nonlinear.
Step-by-step explanation:
ANEXO 2
Identifica los objetos con los que se mide la masa y el volumen, y escribe en donde corresponda.
Manómetro
VOLUMEN
MASA
Pipetas
Fórmula de densidad,
Probetas
Báscula.
Matraz
Balanzas
Fórmula volumen
Vaso de precipitación
The objects used to measure mass are Balances and Scales. The objects used to measure Volume are Manometer, Pipettes, Graduated cylinders, Flasks, Volumetric flasks and Beakers. Here Density formula can be used to measure both mass and volume.
The problem is asking to match different measuring tools with the measurements they are used for, i.e., mass or volume.
The first tool is a manometer. A manometer is used to measure pressure and not mass or volume, so it does not belong in either category.
The next set of tools are pipettes, graduated cylinders, and volumetric flasks. These tools are all used to measure volume, so they belong in the volume category.
The next set of tools are scales and balances. These tools are used to measure mass, so they belong in the mass category.
The formula for density can be used to calculate the mass of an object given its volume and density, or the volume of an object given its mass and density, so it belongs in both categories.
Finally, a beaker or a graduated cylinder can be used to measure volume, so it belongs in the volume category.
Therefore, the correct categorization of the measuring tools are as follows
Volume
Pipettes
Graduated cylinders
Volumetric flasks
Beaker or graduated cylinder
Mass
Scales
balances
Both
Formula for density
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The temperature at any point (x, y) in a steel plate is T = 900 − 0.7x2 − 1.3y2, where x and y are measured in meters. At the point (3, 9), find the rates of change of the temperature with respect to the distances moved along the plate in the directions of the x- and y-axes.
At point (3, 9), the rate of change of temperature with respect to the x-axis is -4.2 °C/m, and with respect to the y-axis, it is -23.4 °C/m.
To find the rates of change of the temperature with respect to the distances moved along the x- and y-axes at point (3, 9), you need to compute the partial derivatives of the temperature function T(x, y) = 900 - 0.7x^2 - 1.3y^2 with respect to x and y.
For the x-axis:
∂T/∂x = -1.4x
At point (3, 9), ∂T/∂x = -1.4(3) = -4.2 °C/m
For the y-axis:
∂T/∂y = -2.6y
At point (3, 9), ∂T/∂y = -2.6(9) = -23.4 °C/m
So, at point (3, 9), the rate of change of temperature with respect to the x-axis is -4.2 °C/m, and with respect to the y-axis, it is -23.4 °C/m.
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Determine the equation of the directrix of r = 26. 4/4 + 4. 4 cos(theta) A. X = -6 B. Y = 6 C. X = 6
The equation of the directrix is X = 6 (Option C).
To determine the equation of the directrix of the polar equation r = 26.4/(4 + 4.4cos(theta)), we need to find the constant value of either x or y. This equation is in the form r = ed/(1 + ecos(theta)), where e is the eccentricity, and d is the distance from the pole to the directrix.
In our case, 26.4 = ed and 4.4 = e. To find the value of d, we can divide 26.4 by 4.4:
d = 26.4 / 4.4 = 6
Since the directrix is a vertical line, it has the form x = constant. In this case, the constant is 6.
So, the equation of the directrix is X = 6 (Option C).
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f(x)=1/2x^4+2x^3 is concave up when f”(x) is
The function f(x) = (¹/₂)x⁴ +2x³ is concave up when f''(x) > 0, which is true when x > 0 or x < -2.
What is the concavity of the function?The concavity of a function is determined by taking the second derivative.
f'(x) = 2x³ + 6x²
f''(x) = 6x² + 12x
To find out when f(x) is concave up, we need to determine when f''(x) is positive;
f''(x) > 0
6x² + 12x > 0
6x(x + 2) > 0
When x > 0, both factors are positive, and the inequality is true.
When x < -2, both factors are negative, and the inequality is true.
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The spokes on a bicycle wheel divide the wheel into congruent sections. What is the measure of each arc in this circle?
The measure of each arc in the circle is given by: 360 degrees / n
where n= number of spokes
If the spokes on a bicycle wheel divide the wheel into congruent sections, then each section is an equal angle at the center of the circle. Since there are "n" spokes on the wheel, the circle will be divided into "n" congruent sections.
Therefore, the measure of each arc in the circle is given by:
= 360 degrees / n
For example, if there are 18 spokes on the wheel, then each arc will have a measure of:
360 degrees / 18 = 20 degrees
So each arc would measure 20 degrees.
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The table shows the blood pressure of 16 clinic patients.what is the interquartile range of the data? a)7.75 b)8.50 c)9.25 d)10.75
The closest option to this value is d) 10.75, but none of the options is an exact match.
To find the interquartile range (IQR) of the data, we need to first find the first quartile (Q1) and the third quartile (Q3).
To do this, we can arrange the data in order from smallest to largest:
98, 100, 104, 105, 106, 110, 112, 115, 116, 118, 120, 122, 126, 130, 136, 140
The median of the data is the average of the two middle values, which are 112 and 115. So, the median is (112 + 115) / 2 = 113.5.
To find Q1, we need to find the median of the data values below the median. These are:
98, 100, 104, 105, 106, 110, 112, 115
The median of these values is (106 + 110) / 2 = 108.
To find Q3, we need to find the median of the data values above the median. These are:
116, 118, 120, 122, 126, 130, 136, 140
The median of these values is (122 + 126) / 2 = 124.
Now we can calculate the interquartile range (IQR) as the difference between Q3 and Q1:
IQR = Q3 - Q1 = 124 - 108 = 16.
Therefore, the interquartile range of the data is 16, or in decimals 16.00.
The closest option to this value is d) 10.75, but none of the options is an exact match.
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Why is the quotient of three divided by one-fifth different from the quotient of one-fifth divided by three? Tell a story that could describe each situation. I don't know how to word it, please help. Please also give me the sums
The order of division affects the result; 3 ÷ 1/5 is 15 and 1/5 ÷ 3 is 1/15.
How are the quotients different?To find the answer, we can calculate the quotient of three divided by one-fifth, which is:
3 ÷ (1/5) = 15
And the quotient of one-fifth divided by three is:
(1/5) ÷ 3 = 1/15
These two quotients are different because the order of division changes the result. In the first case, we divide 3 by a smaller number (one-fifth), which results in a larger quotient (15). In the second case, we divide a smaller number (one-fifth) by a larger number (three), which results in a smaller quotient (1/15).
To give a story describing each situation:
For the first situation, imagine a pizza that is divided into five equal slices, and three hungry friends who want to share it. Each friend gets one-fifth of the pizza, but they want to know how much pizza they would get if they each had three-fifths. To find out, they combine their slices, which gives them three out of the five slices. The total amount of pizza they have is now three-fifths of the pizza, and they can each take one-third of that amount, which is 15% of the original pizza.For the second situation, imagine a group of three friends who want to share a small bag of candy that has five pieces in it. Each friend gets one-fifth of the candy, but they want to know how much candy they would get if they each had three pieces. To find out, they divide the total number of pieces (five) by the number of friends (three), which gives them one and two-thirds pieces each, or one-fifteenth of the bag.Learn more about quotient
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We want to evaluate the integral X +34 +16 dx, we use the trigonometric substitution X and dx = do and therefore the integrar becomes, in terms or o, de The antiderivative in terms of 8 is (do not forget the absolute value) 1 = + Finally, when we substitute back to the variable x, the antiderivative becomes T Use for the constant of integration
The antiderivative of the given integral is (X^2/2) + 50X + C, where C is the constant of integration.
This is obtained by integrating the given polynomial directly without the need for trigonometric substitution.First, let's rewrite the integral: ∫(X + 34 + 16) dx. Since the integrand is a polynomial, we don't need trigonometric substitution. Instead, we can find the antiderivative directly:
∫(X + 34 + 16) dx = ∫(X + 50) dx.
Now, find the antiderivative:
(X^2/2) + 50X + C, where C is the constant of integration.
So, the antiderivative of ∫(X + 34 + 16) dx is (X^2/2) + 50X + C.
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If the circumference of a circle is 50. 4 ft, what is its area? (Use π = 3. 14) *
2 points
50. 24 sq ft
113. 04 sq ft
202. 24 sq ft
314 sq ft
If the circumference of a circle is 50. 4 ft, its area is 202.24 sq ft. Correct option is C: 202.24 sq ft.
The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. We are given that the circumference is 50.4 ft, so we can solve for the radius:
50.4 = 2πr
r = 50.4 / (2π)
r ≈ 8.02 ft
Now we can use the formula for the area of a circle: A = πr²
A = 3.14 * (8.02)²
A ≈ 202.24 sq ft
Therefore, the answer is option C: 202.24 sq ft.
Alternatively, to find the area of a circle with a circumference of 50.4 ft, we will first find the radius using the formula for circumference (C = 2πr) and then use the formula for the area of a circle (A = πr²). Using π = 3.14:
Solve for the radius (r):
C = 2πr
50.4 = 2(3.14)r
r = 50.4 / (2 * 3.14)
r ≈ 8 ft
Calculate the area (A):
A = πr²
A = 3.14 * (8²)
A = 3.14 * 64
A ≈ 201.06 sq ft
The closest answer among the options provided is 202.24 sq ft. Correct option is C: 202.24 sq ft.
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Consider the following cash flows: year cash flow 0 −$28,500 1 15,200 2 13,700 3 10,100 a. what is the profitability index for the cash flows if the relevant discount rate is 10 percent?
The profitability index is 0.1237.
To find the profitability index (PI), we need to divide the present value of the cash flows by the initial investment.
To calculate the present value of the cash flows, we need to discount each cash flow to its present value and then add them up. Using a discount rate of 10%, we get:
Year 0: -$28,500 / [tex](1 + 0.10)^0[/tex]= -$28,500
Year 1: $15,200 /[tex](1 + 0.10)^1[/tex]= $13,818.18
Year 2: $13,700 / [tex](1 + 0.10)^2[/tex] = $10,881.68
Year 3: $10,100 /[tex](1 + 0.10)^3[/tex] = $7,322.51
The sum of the present values is:
PV = -$28,500 + $13,818.18 + $10,881.68 + $7,322.51 =
PV = $3,521.37
The profitability index is therefore:
PI = PV / Initial Investment = $3,521.37 / $28,500 = 0.1237
So the profitability index is 0.1237.
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The perimeter of a semicircle is 35. 98 millimeters. What is the semicircle's radius Use 3. 14 for a. Millimeters Submit explain
If the perimeter of a semicircle is 35. 98 millimeters, 7 mm is the semicircle's radius.
A semi-circle refers to half of the circle. The circle is cut along the diameter to form a semi-circle.
A diameter is a line segment that passes through the center of the circle and touches the boundary of the circle from both ends.
The perimeter of the semi-circle is the sum of the length of the diameter and the circumference of the semi-circle.
P = 2r + πr
where P is the perimeter
r is the radius
P = 35.98 mm
35.96 = 2r + 3.14r
35.96 = 5.14r
r = 7 mm.
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In your pocket you have 4 ones, 2 fives, and a twenty dollar bill. What is the probability of picking out the twenty?
The probability of picking a 20 dollar bill is 1/7 or 14.3%
How do we calculate for the probability of picking up a 20 dollar bill?The probability of a thing is the likelihood or number of chances that such a thing will occur. For the scenario given,
There are a total of 7 bills in your pocket
1, 1, 1, 1,
5, 5,
20.
To find the probability of picking out the twenty dollar bill, divide the number of twenty dollar bills by the the total of the number of bills you are with.
Probability = 1/ 7 which can be converted to % = 14.3%.
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Find the side x, giving answer to 1 decimal place
Answer:
Set your calculator to degree mode.
Using the Law of Sines:
7/sin(40°) = x/sin(81°)
x = 7sin(81°)/sin(40°) = 10.8
Answer:
10.8=x
Step-by-step explanation:
Using the Law of Sines, we can put together the fact that
[tex]\frac{sin A}{a} =\frac{sinB}{b}[/tex]
Substitute our given values from the triangle:
[tex]\frac{sin 81}{x} =\frac{sin40}{7}[/tex]
Turn the sines into a decimal:
[tex]\frac{0.9876}{x} =\frac{0.6427}{7}\\[/tex]
cross multiply using butterfly method
0.988·7=0.643x
solve for x
6.916=0.643x
divide both sides by 0.643
10.8=x (round to nearest tenth)
Hope this helps! :)
100 POINTS!!!! PLEASE HELP!! ITS DUE IN 1 HOUR!!!!!!!!!!!!!!
An object moves in simple harmonic motion with period 8 minutes and amplitude 12m. At time =t0 minutes, its displacement d from rest is −12m, and initially it moves in a positive direction.
Give the equation modeling the displacement d as a function of time t
An object moves in simple harmonic motion with a period of 8 minutes and an amplitude of 12 m. At time =t0 minutes, its displacement d from rest is −12m, and initially, it moves in a positive direction. We can write the final equation for the displacement d as a function of time t: d(t) = 12 * cos((π/4)t + π)
To model the displacement d as a function of time t for an object in simple harmonic motion with a period of 8 minutes and an amplitude of 12m, we'll use the following equation:
d(t) = A * cos(ωt + φ)
where:
- d(t) is the displacement at time t
- A is the amplitude (12m in this case)
- ω is the angular frequency, calculated as (2π / period)
- t is the time in minutes
- φ is the phase angle, which we'll determine based on the initial conditions
Since the period is 8 minutes, we can calculate the angular frequency as follows:
ω = (2π / 8) = (π / 4)
At t = 0 minutes, the displacement is -12m, and the object moves in a positive direction. So we have:
-12 = 12 * cos(φ)
Dividing both sides by 12:
-1 = cos(φ)
Therefore, φ = π (or 180°) since the cosine of π is -1.
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Consider circle o with diameter lm and chord pq.
if lm = 20 cm, and pq = 16 cm, what is the length of rm, in centimeters?
If circle has diameter lm and chord pq, lm = 20 cm, and pq = 16 cm, the length of RM is 10√2 centimeters.
In a circle, a diameter is a chord that passes through the center of the circle. Therefore, the point where the diameter and the chord intersect, in this case, point R, bisects the chord.
Since LM is a diameter, its length is twice the radius of the circle, which means LM = 2r. Thus, we can find the radius of the circle by dividing the diameter by 2: r = LM/2 = 20/2 = 10 cm.
Since point R bisects the chord PQ, RP = RQ = 8 cm (half of PQ). Thus, we need to find the length of RM. To do that, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle RLM with RM as the hypotenuse, so we can use the Pythagorean theorem as follows:
RM² = RL² + LM²
RM² = (10)² + (10)²
RM² = 200
RM = √200 = 10√2 cm
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Complete question is:
Consider circle o with diameter lm and chord pq.
if lm = 20 cm, and pq = 16 cm, what is the length of rm, in centimeters?
If the shapes are scaled copy select a reasonable scale factor that could be applied to shape to 2 create shape 1. i need help asap
A reasonable scale factor that could be applied to Shape 2 to create Shape 1 is 0.5.
What scale factor can be used to transform Shape 2 into Shape 1, if they are scaled copies?When we talk about scaling a shape, we mean changing the size of the shape while maintaining its overall proportions.
This can be done by multiplying all of the dimensions of the shape by the scale factor.
For example, if we wanted to make a shape twice as big, we would multiply all of its dimensions (length, width, and height) by 2. If we wanted to make it half as big, we would multiply all of its dimensions by 0.5.
Looking at the two shapes, we can see that Shape 1 is half the size of Shape 2 in all dimensions.
For example, the height of Shape 1 is half the height of Shape 2, the width of Shape 1 is half the width of Shape 2, and the length of Shape 1 is half the length of Shape 2.
Therefore, to transform Shape 2 into Shape 1, we need to multiply all of its dimensions by 0.5. This will result in a scaled copy of Shape 2 that is identical in shape to Shape 1.
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On the interval [0, 2] the polar curve r = 8o2 has arc length ______ units.
The arc length of the polar curve r = 8θ^2 on the interval [0, 2] is approximately 70.71 units.
The polar curve r = 8θ^2 on the interval [0, 2] has an arc length which can be calculated using the formula for arc length in polar coordinates:
L = ∫√(r^2 + (dr/dθ)^2) dθ, from θ = 0 to θ = 2.
First, we need to find the derivative dr/dθ:
r = 8θ^2, so dr/dθ = 16θ.
Now, plug r and dr/dθ into the arc length formula:
L = ∫√((8θ^2)^2 + (16θ)^2) dθ, from θ = 0 to θ = 2.
Simplify the integrand:
L = ∫√(64θ^4 + 256θ^2) dθ, from θ = 0 to θ = 2.
Factor out 64θ^2:
L = ∫√(64θ^2(1 + θ^2)) dθ, from θ = 0 to θ = 2.
Now, apply the substitution u = 1 + θ^2, so du = 2θ dθ:
L = 32∫√(u) du, from u = 1 to u = 5.
Integrate and evaluate:
L = (32/3)(u^(3/2)) | from u = 1 to u = 5.
L = (32/3)(5^(3/2) - 1^(3/2)).
L ≈ 70.71 units.
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The velocity function is v(t)=−t2+5t−4 for a particle moving along a line. Find the displacement and the distance traveled by the particle during the time interval [-2,5].
displacement =
distance traveled =
Integrate the absolute value of the velocity function over each subinterval, and sum up the results to find the distance traveled.
Remember, displacement is the net change in position, while distance traveled is the total length of the path the particle moves along.
Hi! To find the displacement and distance traveled during the time interval [-2, 5], we need to integrate the velocity function v(t) = -t^2 + 5t - 4 over the given interval.
First, let's find the antiderivative of v(t) which gives us the position function s(t):
s(t) = ∫(-t^2 + 5t - 4) dt = (-1/3)t^3 + (5/2)t^2 - 4t + C
For displacement, we simply need to find the difference in the position function at the endpoints of the interval:
displacement = s(5) - s(-2)
For distance traveled, we need to consider both the positive and negative parts of the velocity function. Find the time when v(t) = 0 to determine when the particle changes direction:
-t^2 + 5t - 4 = 0
Solve this quadratic equation for t. Next, divide the interval [-2, 5] into subintervals based on the values of t where the particle changes direction. Integrate the absolute value of the velocity function over each subinterval, and sum up the results to find the distance traveled.
Remember, displacement is the net change in position, while distance traveled is the total length of the path the particle moves along.
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A watch designer claims that men have wrist breadths with a mean equal to 9 cm. A simple random sample of wrist breadths
of 72 men has a mean of 8.91 cm. The population standard deviation is 0.36 cm.
Assume a confidence level of a = 0.01. Find the value of the test statistic z using
formula below
Z=
X-H
σ
O2.12
O-1.27
O 0.06
O-2.12
The value of the test statistic z is approximately -2.12. The Option D is correct.
What is the value of the test statistic z?To test the hypothesis that the mean wrist breadth of men is equal to 9 cm, we will use a one-sample z-test.
The null hypothesis is: H0: µ = 9 cm
The alternative hypothesis is: Ha: µ ≠ 9 cm
We are given a sample of n = 72 men with a sample mean of x = 8.91 cm and a population standard deviation = 0.36 cm.
The test statistic for a one-sample z-test is given by: z = (x - µ) / (o / sqrt(n))
Substituting the given values, we get:
= (8.91 - 9) / (0.36 / sqrt(72))
z = -2.119
At a significance level of a = 0.01, the critical values for a two-tailed test are ±2.576.
Since our test statistic (-2.119) falls outside of this range, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean wrist breadth of men is not equal to 9 cm.
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Quickly please anyone
f(x) = 6x² - 3x + ²
2
X
f(-2) = [?]
Be sure to simplify your answer.
Answer:
Ans=28
Step-by-step explanation:
ƒ(x) = 6x2 - 3x + 22xf( - 2)=[?]
at ƒ(-2)
Substitute each x with -2
ƒ(-2) = 6(-2)2 - 3(-2) - 2
ƒ(-2) = 6(4) - 3(-2) - 2
ƒ(-2) = 24 + 6 + 0 - 2
ƒ(-2) = 28
I hope I was right
Pls help
a polynomial function is represented by the data in the table
x 0 i 1 i 2 i 3 i 4 i
f(x) -24 i -21¾ i -14 i ¾ i 24 i
choose the function represented by the data.
1. f(x) = x3 − x2 − 24
2. f(x) [tex]\frac{x}{4}^{3}[/tex] + 2[tex]x^{2}[/tex] -24
3. f(x)= -2[tex]\frac{1}{4} x^{2}[/tex] + 24
4. f(x)= [tex]\frac{3}{4} x^{2}[/tex] -3x + 24
The function represented by the data is f(1/4)x³ + 2x² - 24. The correct option is 2.
In the given table, we have the values of x and f(x) for x=0,1,2,3, and 4. We need to find a polynomial function that satisfies these data points.
Looking at the table, we can see that f(x) is negative for x=0,1,2 and positive for x=3,4. This suggests that the polynomial has a root or a zero between x=2 and x=3.
To find the degree of the polynomial, we count the number of data points given. Since we have 5 data points, we need a polynomial of degree 4.
We can use interpolation to find the coefficients of the polynomial. One way to do this is to set up a system of equations using the data points:
f(0) = -24 = a(0)⁴ + b(0)³ + c(0)² + d(0) + e
f(1) = -21.75 = a(1)⁴ + b(1)³ + c(1)² + d(1) + e
f(2) = -14 = a(2)⁴ + b(2)³ + c(2)² + d(2) + e
f(3) = 0.75 = a(3)⁴ + b(3)³ + c(3)² + d(3) + e
f(4) = 24 = a(4)⁴ + b(4)³ + c(4)² + d(4) + e
Solving this system of equations gives us the polynomial function:
f(x) = -0.25x⁴ + 2x³ - 2.75x² - 0.5x + 24
Therefore, the correct option is 2. f(x) = (1/4)x³ + 2x² - 24.
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How would I solve this equation by factoring m²-64 = 0
The force on a particle is described by 8x^3-5 at a point s along the z-axis. Find the work done in moving the particle from the origin to x = 4.
The work done in moving the particle from the origin to x = 4 under the influence of the force F (x) = 8[tex]x^3[/tex]-5 is 492 units of work.
The work done in moving a particle along a path under the influence of a force, we use the work-energy principle.
This principle states that the work done on a particle by a force is equal to the change in the particle's kinetic energy.
Mathematically this can be expressed as:
W = ΔK
Where
W is the work done,
ΔK is the change in kinetic energy and
Both are scalar quantities.
The work done by a force on a particle along a path is given by the line integral:
W = ∫ C F · ds
Where,
C is the path,
F is the force,
ds is the differential displacement along the path and denotes the dot product.
In the case where the force is a function of position only (i.e., F = F(x,y,z)), we can evaluate the line integral using the parametric equations for the path.
If the path is given by the parameterization r(t) = <x(t), y(t), z(t)>, then we have:
W = ∫ [tex]a^b[/tex] F(r(t)) · r'(t) dt
The work done in moving the particle from the origin to a final position at x = 4. We can evaluate the work done using the definite integral of the force from x = 0 to x = 4, as shown in the solution.
The initial kinetic energy is zero.
The work done by the force in moving the particle from x = 0 to x = 4 is given by the definite integral:
W = ∫ F(x) dx
Substituting the given expression for the force, we have:
W = ∫0 (8x - 5) dx
Integrating with respect to x, we have:
W = [(2x - 5x)]_0
W = (2(4) - 5(4)) - (2(0) - 5(0))
W = 512 - 20
W = 492
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FOIL the equation, don't need to solve!
(2x-1)(x+2)
When we multiply (2x - 1) and (x + 2) using FOIL method, we get:
(2x - 1)(x + 2) = 2x(x) + 2x(2) - 1(x) - 1(2)
= 2x² + 4x - x - 2
= 2x² + 3x - 2
Therefore, the product of (2x - 1) and (x + 2) is 2x² + 3x - 2.
