The absolute maximum value of f(x) on the interval [-2,3] is 201 and occurs at x = -2, and the absolute minimum value of f(x) on the interval [-2,3] is -79 and occurs at x = 2.
To find the absolute maximum and absolute minimum values of f(x) on the interval [-2,3], we need to first find the critical points of the function and evaluate the function at these points as well as at the endpoints of the interval.
To find the critical points, we need to find where the derivative of the function equals zero or does not exist. Taking the derivative of f(x), we get:
f'(x) = 12x^3 - 12x^2 - 24x
Setting this equal to zero and factoring out 12x, we get:
12x(x^2 - x - 2) = 0
Using the quadratic formula to solve for x^2 - x - 2 = 0, we get:
x = -1, 0, 2
These are our critical points.
Now we evaluate f(x) at the critical points and the endpoints of the interval:
f(-2) = 201
f(-1) = 6
f(0) = 1
f(2) = -79
f(3) = 16
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$1,000 is deposited into a savings account. Interest is compounded annually. After 1 year, the value of the account is $1,020. After 2 years, the value of the account is $1,040. 40. This scenario can be represented by an exponential function of the form fx=1000bx, where fxis the amount in the savings account, and x is time in years. What is the value of b?
The value of b in the exponential function fx =1000bx is 1.02.
The problem states that interest is compounded annually, which means that the interest earned in a year is added to the principal amount at the end of the year. Using the given information, we can set up the following equations:
f₁ = 1000(1+b) = 1020
f₂ = 1000(1+b)² = 1040.40
We can solve for b by dividing the second equation by the first equation and taking the square root:
(1+b)² / (1+b) = 1040.40 / 1020
1+b = √1.02
b = 1.02 - 1 = 0.02
Therefore, the value of b is 0.02 or 2%. The exponential function is fx = 1000(1+0.02)ᵗ, where t is the time in years.
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Mr robins earns a commission on each airfare he books. At the end of the day he had booked 208. 60 worth of airfare and earned 31. 29
Mr. Robins earns a commission of 15% on the airfares he books, as he earned $31.29 on $208.60 worth of airfare bookings.
Let x be the amount of commission earned by Mr. Robins on the airfares he booked. Then, we can write the equation:
x = 15% of $208.60
Simplifying this equation, we get:
x = 0.15 x $208.60
x = $31.29
Therefore, Mr. Robins earned a commission of $31.29 on $208.60 worth of airfare bookings. To verify this, we can calculate his commission rate as:
Commission rate = Commission earned / Airfare bookings
Commission rate = $31.29 / $208.60
Commission rate = 0.15 or 15%
Hence, Mr. Robins earns a commission of 15% on the airfares he books.
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Q5. Compute the trapezoidal approximation for | Vx dx using a regular partition with n=6.
The trapezoidal approximation for | Vx dx using a regular partition with n=6 is approximately 0.1901.
How to find the trapezoidal approximation for a function?To compute the trapezoidal approximation for | Vx dx using a regular partition with n=6, we can use the formula:
Tn = (b-a)/n * [f(a)/2 + f(x1) + f(x2) + ... + f(xn-1) + f(b)/2]
where Tn is the trapezoidal approximation, n=6 is the number of partitions, a and b are the limits of integration, and x1, x2, ..., xn-1 are the partition points.
In this case, we have | Vx dx as the function to integrate. Since there are no given limits of integration, we can assume them to be 0 and 1 for simplicity.
So, a=0 and b=1, and we need to find the values of f(x) at x=0, 1/6, 2/6, 3/6, 4/6, and 5/6 to use in the formula.
We can calculate these values as follows:
f(0) = | V0 dx = 0
f(1/6) = | V1/6 dx = V(1/6) - V(0) = sqrt(1/6) - 0 = 0.4082
f(2/6) = | V2/6 dx = V(2/6) - V(1/6) = sqrt(2/6) - sqrt(1/6) = 0.2317
f(3/6) = | V3/6 dx = V(3/6) - V(2/6) = sqrt(3/6) - sqrt(2/6) = 0.1547
f(4/6) = | V4/6 dx = V(4/6) - V(3/6) = sqrt(4/6) - sqrt(3/6) = 0.1104
f(5/6) = | V5/6 dx = V(5/6) - V(4/6) = sqrt(5/6) - sqrt(4/6) = 0.0849
Now we can substitute these values in the formula and simplify:
T6 = (1-0)/6 * [0/2 + 0.4082 + 0.2317 + 0.1547 + 0.1104 + 0.0849/2]
= 0.1901
Therefore, the trapezoidal approximation for | Vx dx using a regular partition with n=6 is approximately 0.1901.
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If α and β are the zeros of x^2-x+k, and 3α+2β=20, find k.
The solution of the given problem of quadratic equation comes out to be K thus has a value of 63/4.
What is quadratic equation?Regression modelling uses the polynomial solutions x = ax² + b + c=0 for one-variable equations. The First Principle of Algebra states that there can only be one solution because it has an extra order. There are both simple and complex solutions available. As the name suggests, a "non-linear formula" has four variables. This implies that there may only be one squared word. In the equation "ax² + bx + c = 0.
Here,
We know that if and are the zeros of the quadratic equation x²-x+k then:
=> α + β = 1
=> αβ = k
Additionally, we are told that 3 + 2 = 20.
We may find as = 1 - by using the equation + = 1.
By replacing this expression for in terms of in the formula k = a, we obtain:
=> (1 - β)β = k
=> β² - β + k = 0
=> 3α + 2(1 - α) = 20
=> α = 6 - 2β/3
=> (6 - 2β/3)²- (6 - 2β/3) + k = 0
=> 4β² - 36β + 72 + 3k = 0
=> 3(6 - 2β/3) + 2β = 20
=> 4β/3 = 2
=> β = 3/2
=> 4(3/2)² - 36(3/2) + 72 + 3k = 0
When we simplify and find k, we obtain:
=>k = 63/4
K thus has a value of 63/4.
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If November 30 falls on a Sunday, then December 25 of that same year falls on which day of the week? (November has 30 days)
Step-by-step explanation:
Three weeks would be the 21st and would be Sunday too, then
22 Mon
23 Tues
24 Wed
25 Thur
Landon was comparing the price of apple juice at two stores. The equation y=0. 96xy=0. 96x represents what Landon would pay in dollars and cents, yy, for xx bottles of apple juice at store A. Landon can buy 14 bottles of apple juice at Store B for a total cost of $34. 16
The equation y=0.96x represents the cost in dollars and cents, y, for x bottles of apple juice at store A. This means that if Landon wants to buy x bottles of apple juice from store A, he would pay 0.96x dollars and cents. However, we do not know the value of x from the given information.
On the other hand, we know that Landon can buy 14 bottles of apple juice from store B for $34.16. This means that the cost of one bottle of apple juice at store B is $2.44 (34.16 ÷ 14). We do not know the cost of one bottle of apple juice at store A, but we can use the equation y=0.96x to find out.
Let's assume that Landon wants to buy 14 bottles of apple juice from store A as well. We can substitute x=14 in the equation to find the total cost:
y = 0.96(14) = 13.44
This means that Landon would pay $13.44 for 14 bottles of apple juice at store A. However, we can see that buying 14 bottles of apple juice from store B is cheaper than buying the same amount from store A, as Landon would pay $34.16 at store B and $13.44 at store A. Therefore, it is more cost-effective for Landon to buy apple juice from store B in this case.
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using known taylor series find the first 4 nonzero terms of thetaylor series for the function f(t)=e^(t)cos(t) about 0
The first four nonzero terms are 1 + t - (t^2)/2 - (t^3)/3 + (t^4)/8
To find the first 4 nonzero terms of the Taylor series for the function f(t) = e^(t)cos(t) about 0,
we can use the known Taylor series for e^(t) and cos(t).
Taylor series:
The Taylor series for e^(t) is:
e^(t) = 1 + t + (t^2)/2! + (t^3)/3! + ...
And the Taylor series for cos(t) is:
cos(t) = 1 - (t^2)/2! + (t^4)/4! - (t^6)/6! + ...
To find the Taylor series for f(t) = e^(t)cos(t), we can multiply these two series together using the distributive property of multiplication. We get:
f(t) = (1 + t + (t^2)/2! + (t^3)/3! + ...) * (1 - (t^2)/2! + (t^4)/4! - (t^6)/6! + ...)
Expanding this out, we get:
f(t) = 1 + t - (t^2)/2 - (t^3)/3 + (t^4)/8 + (t^5)/15 - (t^6)/72 - ...
The first 4 nonzero terms of this series are:
f(t) = 1 + t - (t^2)/2 - (t^3)/3 + (t^4)/8 + ...
So, the first 4 nonzero terms of the Taylor series for f(t) = e^(t)cos(t) about 0 are:
1 + t - (t^2)/2 - (t^3)/3 + (t^4)/8
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Julia works at a music store. One of her jobs is to stock new CDs on the shelf. A recent order arrived with 215 classical CDs, 125 jazz CDs, and 330 soft rock CDs. How many groups will Julia use to arrange all of the CDs?
Julia will use 10 groups to arrange all of the CDs.
To determine the number of groups Julia will use to arrange all of the CDs, we need to find the greatest common divisor of the numbers 215, 125, and 330.
First, we can check if any of the numbers are divisible by 5:
215 is not divisible by 5
125 is divisible by 5 (125 ÷ 5 = 25)
330 is divisible by 5 (330 ÷ 5 = 66)
Now we divide 125 and 330 by 5:
125 ÷ 5 = 25
330 ÷ 5 = 66
Next, we check if any of the numbers are divisible by 2:
25 is not divisible by 2
66 is divisible by 2 (66 ÷ 2 = 33)
Now we divide 66 by 2:
66 ÷ 2 = 33
Therefore, the greatest common divisor of 215, 125, and 330 is 5 × 2 = 10.
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we introduced wind chill as a way of calculating the apparent temperature a person would feel as a function of the real air temperature, I, and V in
mph. Then the wind chill (i.e., the apparent temperature) is:
W(T, V) = (35.74 + 0.6215T - 35.75V^0.16) / 0.4275TV^0.16
(a) By calculating the appropriate partial derivative, show that
increasing T always increases W. (
b) Under what conditions does increasing V decrease W? Your
answer will take the form of an inequality involving T.
(c) Assuming that W should always decrease when V is in- creased, use your answer from (b) to determine the largest domain in which this formula for W can be used.
a) The partial derivative of W with respect to T is always positive, which means that increasing T always increases W.
b) Increasing V decreases W if V is greater than
[tex]((0.8T - 0.6215) / 5.71)^{(1/0.16)} .[/tex]
c) The largest domain in which the inequality derived in (b) holds true is:
T > 0.7769. This means that the wind chill formula can be used only for
air temperatures above 0.7769 degrees Fahrenheit.
(a) To show that increasing T always increases W, we need to calculate the partial derivative of W with respect to T and show that it is always positive.
∂W/∂T = [tex]0.6215/0.4275V^{0.16} - (35.75V^{0.16})/0.4275TV^{0.16}^{2}[/tex]
Simplifying this expression, we get:
∂W/∂T = [tex]1.44(0.6215 - 0.0275V^{0.16T}) / V^{0.16}T^{2}[/tex]
Since 1.44 and[tex]V^{0}.16T^{2}[/tex] are always positive, the sign of the partial derivative depends on the sign of[tex](0.6215 - 0.0275V^{0.16T} ).[/tex]
Since 0.0275 is always positive and [tex]V^{0.16T}[/tex] is also always positive, we see that [tex](0.6215 - 0.0275V^{0.16T} )[/tex] is always positive.
(b) To find the conditions under which increasing V decreases W, we need to calculate the partial derivative of W with respect to V and show that it is always negative.
∂W/∂V = [tex](-35.750.16V^{(-0.84)} (35.74+0.6215T-35.75V^{0.16} )-0.6215V^{(-0.16} ))/0.4275TV^{(0.16)}[/tex]
Simplifying this expression, we get:
∂W/∂V = [tex]-0.16(0.6215+5.71V^{0.16-0.8T} ) / TV^{0.84}[/tex]
The sign of the partial derivative depends on the sign of [tex](0.6215+5.71V^{0.16-0.8T} ).[/tex]
If [tex]0.6215+5.71V^{0.16-0.8T} < 0[/tex], then the partial derivative is negative and increasing V decreases W.
Solving this inequality for V, we get:
[tex]V > ((0.8T - 0.6215) / 5.71)^{(1/0.16)}[/tex]
(c) Assuming that W should always decrease when V is increased, we need to find the largest domain in which the inequality derived in (b) holds true.
Since the expression inside the parentheses must be positive for a real solution, we have:
0.8T - 0.6215 > 0
T > 0.7769
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Marsha threw her math book off a 30 foot building. The equation of the book can be represented by the equation h=-16[tex]x^{2}[/tex]+24x+30. What is the maximum height
of Marsha's math book?
The maximum height of Marsha's math book is 36 feet.
To find the maximum height of Marsha's math book, we need to find the vertex of the parabolic equation h = [tex]-16x^2 + 24x + 30[/tex]. The vertex of a parabola is the highest or lowest point on the curve, depending on whether the parabola opens upward or downward.
To find the x-coordinate of the vertex, we can use the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation [tex]ax^2 + bx + c[/tex]. In this case, a = -16 and b = 24, so we have:
x = -b/2a = -24/(2*(-16)) = 0.75
To find the y-coordinate of the vertex, we can substitute x = 0.75 into the equation h = [tex]-16x^2 + 24x + 30[/tex], which gives us:
h = [tex]-16(0.75)^2 + 24(0.75) + 30 = 36[/tex]
Therefore, the maximum height of Marsha's math book is 36 feet.
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Use the following bond listing for Pacific Bell to answer the following: A 5-column table with 1 row. Column 1 is labeled Bonds with entry PacBell 6 and StartFraction 5 Over 8 EndFraction 34. Column 2 is labeled current yield with entry 6. 55. Column 3 is labeled Volume with entry 5. Column 4 is labeled Close with entry 99 and one-fourth. Column 5 is labeled net change with entry + StartFraction 1 Over 8 EndFraction. How many bonds were traded during this session?
5 bonds were traded during this session.
Based on the provided bond listing for Pacific Bell, the number of bonds traded during this session is 5. Here's the breakdown of the information in the 5-column table:
- Column 1 (Bonds): PacBell 6 5/8 34
- Column 2 (Current Yield): 6.55
- Column 3 (Volume): 5
- Column 4 (Close): 99 1/4
- Column 5 (Net Change): +1/8
The "Volume" column indicates the number of bonds traded during the session. In this case, the volume entry is 5. Therefore, 5 bonds were traded during this session.
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HELP!! I need the answer to pass 10th grade and im stumped D:
AB is dilated by a scale factor of 3 to form A'B'. Point O, which lies on AB, is the center of dilation.
The slope of AB is 3. The slope of A'B' is 3. A'B' passes through point O.
What is dilation in mathematics?Dilation is a process of transformation used to resize an object.
The items are enlarged or shrunk through dilation. An image that retains the original shape is created by this alteration. The size of the form does differ, though.
By multiplying the x and y coordinates of the original figure by the scale factor, you may locate locations on the dilated image when a dilation in the coordinate plane has the origin as the center of dilation.
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If a woman making $29,000 a year receives a cost-of-living increase of 2. 6%, what will her new salary be?
To find the new salary after a 2.6% increase, we need to add 2.6% of the original salary to the original salary.
2.6% of $29,000 can be calculated as:
(2.6/100) x $29,000 = $754
Therefore, the new salary will be:
$29,000 + $754 = $29,754
So the woman's new salary will be $29,754.
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A foam cylinder, with a diameter of 3 inches and height of 4 inches, is carved into the shape of a cone. what is the
maximum volume of a cone that can be carved? round your answer to the hundredths place.
The maximum volume of a cone that can be carved from the foam cylinder is approximately 9.42 cubic inches.
Given data:
diameter = 3 inches
radius = r = 3 ÷ 2 = 1.5 inches
height = 4 inches
We need to find the maximum volume of a cone that can be carved from the foam cylinder. The volume of a cone is given by the formula:
V = [tex]\frac{1}{3}\pi r^2h[/tex]
where:
V = volume
r = radius of the base
h = height
π = 3.14.
Substituting the r, h, and π values in the formula, we get:
V = [tex]\frac{1}{3}[/tex]π[tex]r^2[/tex]h
V = [tex]\frac{1}{3}[/tex] × π × (1.5)² ×(4)
V = [tex]\frac{1}{3}[/tex] × π × 2.25 ×(4)
V = 3 π
V = 9.42 cubic inches
Therefore, the maximum volume of a cone is 9.42 cubic inches.
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Question 5 of 5
nguyen has the following cans of soup in his pantry:
•4 cans of chicken noodle soup
• 2 cans of tomato soup
• 3 cans of vegetable soup
•3 cans of potato soup
he randomly chooses a can of soup for lunch. what is the probability that he will choose chicken noodle soup?
a. 1/2
b. 1/4
c. 1/6
d. 1/4
please explain how you got the answer as well
The probability that Nguyen will choose a can of chicken noodle soup is 1/3. Therefore, the correct option is B.
To find the probability, you need to divide the number of favorable outcomes (chicken noodle soup cans) by the total number of possible outcomes (total cans of soup). Hence,
1. Count the total number of cans of soup: 4 chicken noodle + 2 tomato + 3 vegetable + 3 potato = 12 cans in total.
2. Count the number of chicken noodle soup cans: 4 cans.
3. Divide the number of chicken noodle soup cans (4) by the total number of cans (12): 4/12.
4. Simplify the fraction: 4/12 can be simplified to 1/3.
Therefore, the probability of choosing a chicken noodle soup is option B: 1/3.
Note: The question is incomplete. The complete question probably is: Nguyen has the following cans of soup in his pantry: 4 cans of chicken noodle soup; 2 cans of tomato soup; 3 cans of vegetable soup; 3 cans of potato soup. He randomly chooses a can of soup for lunch. What is the probability that he will choose chicken noodle soup? a. ½ b. 1/3 c. 1/6 d. ¼.
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In the diagram below, congruent figures 1, 2 and 3 are drawn.
Which sequence of transformations maps figure 1 onto figure 2 and then figure 2 onto figure 3
A sequence of transformations that maps figure 1 onto figure 2 and then figure 2 onto figure 3 include the following: D. a translation followed by a rotation.
What is a translation?In Mathematics and Geometry, a translation can be defined as a type of rigid transformation which moves every point of the object in the same direction, as well as for the same distance.
This ultimately implies that, a translation is a type of rigid transformation that does not change the orientation of the original geometric figure (pre-image).
What is a rotation?In Mathematics, a rotation is a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.
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Complete Question:
Which sequence of transformations maps figure 1 onto figure 2 and then figure 2 onto figure 3?
a reflection followed by a translation
a rotation followed by a translation
a translation followed by a reflection
a translation followed by a rotation
You bought a laptop computer for $525 on the "12 months is the same as cash" plan. The terms of the plan on the contract stated that if
not paid within 12 months, you would be assessed 15. 5 percent APR for the amount on the first day of the plan
If you pay the laptop in 11 months, how much will you have paid?
a. $525
b. $540. 50
c. $595. 50
d. $606. 38
Your answer: a. $525
The "12 months is the same as cash" plan means that if you pay off the laptop within 12 months, you won't be charged any interest.
Since you plan to pay off the laptop in 11 months, which is within the 12-month period, you will not be assessed the 15.5 percent APR.
Therefore, you only need to pay the original cost of the laptop, which is $525.
To summarize, as long as you pay the full amount within the specified 12-month period, you avoid the additional interest charges. In this case, you will pay the laptop off in 11 months, so your total payment will be $525.
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A vase in the shape of a cylinder has a radius of 4. 3 cm and a volume of 1330. 2 cm³ what is the height of the base in centimeters round to the nearest 10th
As per the given values, the height of the vase is approximately 7.3 cm.
The radius of the vase = 4.3cm
The volume of vase = 1330. 2 cm³
Two parallel circular bases are connected by a curving surface to form the three-dimensional object known as a cylinder. There are two round flat sides, two curved edges, and one curved surface.
Using the formula for the volume of a cylinder -
V = πr²h,
where r is the radius and h is the height.
Substituting the values -
1330.2 = π(4.3)²h
1330.2 = 58.09πh
Dividing both sides by 58.09π
1330.2/58.09π = 58.09πh/58.09π
h = 7.27
= 7.3 ( After rounding)
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If you were to randomly survey 20 people at 50 random high schools would this be a random sample? Why or why not?
Yes, if you were to randomly survey 20 people at 50 random high schools, it would be considered a random sample because the process involves randomly selecting people from randomly selected high schools, which prevents selection bias..
A random sample is a subset of a population in which every individual has an equal chance of being selected. In this case, the population is the students at the high schools.
By randomly selecting the 50 high schools, you ensure that each school has an equal opportunity to be part of the sample. This helps to prevent selection bias, as no specific schools are deliberately chosen. Moreover, by surveying 20 random people within each selected school, you further eliminate bias, as each student at the school has an equal chance of being selected for the survey.
This random sampling method is beneficial because it helps to obtain a more representative sample of the larger population of high school students. By including diverse schools and students, the survey results can provide more accurate and generalizable insights.
However, it is important to note that even with random sampling, there may still be some limitations, such as sampling error or non-response bias. To minimize these, it is essential to ensure that the sample size is large enough and that survey procedures are properly executed.
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2x + y = 7
3x - 2y = -7
Answer: x = 1, y = 5
Step-by-step explanation:
I assume you want to solve this system of linear equations:
from the first one:
2x + y = 7
.: y = 7 - 2x
substituting this for the y in the second equation:
3x - 2(7 - 2x) = -7
3x - 14 + 4x = -7
7x = 7
x = 1
From before we know that y = 7 - 2x
so now that we know x = 1, we can say y = 7 - 2(1) = 5
So x = 1, y = 5
To find the quotient of 4. 082 and 10,000, move the decimal point in 4. 082
Choose.
places to the
Choose.
The Quotient of 4.082 and 10,000 is 0.0004082.
Find the qoutient of 4. 082 and 10,000?
To find the quotient of 4.082 and 10,000, we need to move the decimal point in 4.082 four places to the left, since there are four zeros in 10,000.
So, we get:
4.082 ÷ 10,000 = 0.0004082 is the answer.
Explanation.
To find the quotient of 4.082 and 10,000, we need to divide 4.082 by 10,000. When we divide by a number that is a power of 10, we can simplify the calculation by moving the decimal point to the left as many places as there are zeros in the divisor.
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Pls help me with this!! 5 pts and brainliest included for the one who answers first!
Answer:
Step-by-step explanation:
Take the natural logarithm of both sides of the equation to remove the variable from the exponent. ln(e−6w)=ln(952) ln ( e - 6 w ) = ln ( 95 2 ).
Yellowstone national park is a popular field trip destination. this year the
senior class at high school a and the senior class at high school b both
planned trips there. the senior class at high school a rented and filled 2
vans and 8 buses with 254 students. high school b rented and filled 6
vans and 11 buses with 398 students. every van had the same number of
students in it as did the buses. find the number of students in each van and
in each bus
let x represent high school a let y represent high school b
The number of students in each bus is 15, and the number of students in each van is 28.
To find the number of students in each van and bus for the field trip to Yellowstone National Park, we can set up a system of equations using the given information. Let x represent the number of students in each van and y represent the number of students in each bus.
For high school A, we have:
2x + 8y = 254
For high school B, we have:
6x + 11y = 398
Now, we can solve this system of equations using the substitution or elimination method. We will use the elimination method:
Step 1: Multiply the first equation by 3 to make the coefficients of x the same in both equations:
6x + 24y = 762
Step 2: Subtract the second equation from the new first equation:
(6x + 24y) - (6x + 11y) = 762 - 398
13y = 364
Step 3: Divide both sides by 13 to find the value of y:
y = 364 / 13
y = 28
Now that we have the number of students in each bus, we can find the number of students in each van:
Step 4: Substitute y back into the first equation:
2x + 8(28) = 254
2x + 224 = 254
Step 5: Subtract 224 from both sides to find the value of x:
2x = 30
Step 6: Divide both sides by 2 to find x:
x = 15
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Rewrite the equation by completing the square
Answer:
(x-2.75)^2=4.25
Step-by-step explanation:
2x^2-11x+14=0
divide through by two
x^2-5.5x+7=0
x^2-5.5x=-7
x^2-5.5x+(-5.5/2)^2=-7+(-5.5/2)
(x-2.75)^2=4.25
Bill needs a table to display his model train set. the table needs to be 2 times longer and 3 inches shorter
than it is wide and have an area of 4,608 square inches. what does x need to be to fit these requirements?
2x-3
2x - 3 would be 92 - 3 = 89 inches, which is the length of the table
How to find the length?.The table needs to be 2 times longer than it is wide, so its length is 2 times its width, or 2x.
The table also needs to be 3 inches shorter than it is wide, so its width is x + 3 inches.
The area of the table is 4,608 square inches, so we can set up an equation:
2x(x + 3) = 4,608
Simplifying this equation:
2x²+ 6x = 4,608
Dividing both sides by 2:
x²+ 3x - 2,304 = 0
We can solve for x using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = 3, and c = -2,304. Substituting these values:
x = (-3 ± √(3² - 4(1)(-2,304))) / 2(1)
Simplifying:
x = (-3 ± √(9 + 9,216)) / 2
x = (-3 ± √(9,225)) / 2
x = (-3 ± 95) / 2
x = 46 or x = -49
Since the width of the table cannot be negative, we can ignore the negative solution. Therefore, x needs to be 46 inches to fit the given requirements.
The length of the table is 2x, or 2(46) = 92 inches, and the width is x + 3, or 46 + 3 = 49 inches. The area is 92 * 49 = 4,508 square inches, which matches the given area requirement.
So, 2x - 3 would be 92 - 3 = 89 inches, which is the length of the table.
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Solve for 2x+y=10
3x=y
Answer:
x=2
Step-by-step explanation:
2x+y=10
3x=y
2x+3x=10
5x=10
x=2
Find the area of the surface. The part of the plane 4x + 3y + z = 12 that lies inside the cylinder x2 + y2 = 9
The area of the surface is [tex]\sqrt{\frac{15}{4}}\times \pi[/tex] unit square.
To find the area of the surface, we need to first find the intersection curve between the plane and the cylinder.
From the equation of the cylinder, we know that [tex]x^2 + y^2[/tex] = 9200.
We can substitute [tex]x^2 + y^2[/tex] for [tex]r^2[/tex] and rewrite the equation as [tex]r^2[/tex] = 9200.
Next, we can rewrite the equation of the plane as
z = 12 - 4x - 3y.
Now, we can substitute 12 - 4x - 3y for z in the equation [tex]r^2[/tex] = 9200, giving us:
[tex]x^2 + y^2[/tex] = 9200 - [tex](12 - 4x - 3y)^2[/tex]
Expanding and simplifying, we get:
[tex]x^2 + y^2[/tex] = [tex]16x^2 + 24xy + 9y^2 - 24x - 36y + 884[/tex]
Simplifying further, we get:
[tex]15x^2 + 24xy + 8y^2 - 24x - 36y + 884 = 0[/tex]
We can recognize this as the equation of an ellipse:
To find the area of the surface, we need to find the area of this ellipse that lies within the cylinder.
To do this, we can first find the major and minor axes of the ellipse.
We can rewrite the equation as:
[tex]15(x - \frac{4}{5})^2[/tex] + 8([tex]y[/tex] - [tex]\frac{9}{10}[/tex][tex])^{2}[/tex] = 1
So the major axis has length [tex]2/\sqrt{15}[/tex] unit and the minor axis has length [tex]\frac{2}{\sqrt{8} }[/tex] unit.
The area of the ellipse is then given by:
A = π x ([tex]\frac{1}{2}[/tex] x [tex]\frac{2}{\sqrt{15} }[/tex] x ([tex]\frac{1}{2}[/tex] x [tex]\frac{8}{\sqrt{8} }[/tex])
Simplifying we get:
A = π x ([tex]\sqrt{\frac{2}{15} }[/tex]) x ([tex]\sqrt{\frac{2}{8} }[/tex])
A = π x ([tex]\sqrt{\frac{1}{60} }[/tex])
A = [tex]\sqrt{\frac{15}{4}} \times \pi[/tex] unit square
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Your doing practice 4
For a snowboard cost that was reduced by 40% by the end of the season, the snowboard cost $450 when it was new.
How to find original cost?To find the original cost of the snowboard, let the original price of the snowboard be x.
After a 40% reduction in price, the snowboard costs 60% of its original price, therefore the cost remaining percentage of the original prize times the reduction percentage = the price after reduction:
100% - 40% = 60%
60/100 = 0.6
0.6x = 270
Solving for x to get:
x = 270/0.6 = 450
Therefore, the original price of the snowboard was $450.
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An electrical voltage signal is given by the equation V t = + 12sin(5 2), where V is measured in volts and t in milliseconds. Find a general formula that gives all the times when the voltage will be 0. Write your formula in terms of p. (Notice that the answer to this problem is a sequence, not a series. )
A general formula that gives all the times when the voltage will be 0 is t = ±√((pπ)/10)
To find the general formula for the times when the voltage will be 0, we need to analyze the given equation: V(t) = 12sin(5t²). Since V(t) represents the voltage at time t, we want to find the values of t for which V(t) = 0. This will occur when the sine function equals 0.
The sine function, sin(x), is equal to 0 when its argument x is a multiple of π. Mathematically, this can be expressed as:
sin(x) = 0 ⟺ x = nπ, where n is an integer (0, ±1, ±2, ...)
In our case, the argument of the sine function is 5t². Thus, we want to find values of t for which:
5t² = nπ, where n is an integer.
Now, let's solve this equation for t:
t² = (nπ)/5
t = ±√((nπ)/5)
Since the question asks for a formula in terms of p, let's define p as an integer such that p = 2n (n can be any integer). Thus, the formula becomes:
t = ±√((pπ)/10)
This formula represents the general sequence of times t (in milliseconds) when the voltage V(t) will be equal to 0. Here, p is an even integer (0, ±2, ±4, ...) representing different instances when the voltage is zero.
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Solve the problem by integration 6x where x is the distance The force Fin N) applied by a stamping machine in making a certain computer part is F- x2.9.24 (in cm) through which the force acts. Find the work done by the force
To find the work done by the force, we need to integrate the product of the force and the distance over the range of x.
Given that the force is F(x) = x^2 * 9.24 N and the distance is x, we have:
Work = ∫ F(x) * dx
= ∫ (x^2 * 9.24) * dx
= 9.24 ∫ x^2 dx
= 9.24 * [x^3 / 3]
Evaluating the integral between the limits of 0 and 6 (since the distance is given as x), we get:
Work = 9.24 * [(6^3 / 3) - (0^3 / 3)]
= 9.24 * (72)
= 665.28 Joules
Therefore, the work done by the force is 665.28 Joules.
To find the work done by the force, we need to calculate the integral of the force function with respect to distance. Given the force function F(x) = 6x, and the distance x ∈ [0, 2.9], we can set up the integral as follows:
Work = ∫(6x dx) from 0 to 2.9
To find the integral, we'll apply the power rule for integration:
∫(6x dx) = 3x^2 + C
Now, we need to evaluate the definite integral from 0 to 2.9:
Work = (3 * (2.9)^2) - (3 * (0)^2) = 3 * (8.41) = 25.23 N·m
So, the work done by the force is approximately 25.23 N·m.
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