The limit [tex]\lim_{n \to \infty} n(\sum_{i=1}^{n} 9ni^3)[/tex] is equal to ∞.
To find the limit of the given expression, [tex]\lim_{n \to \infty} n(\sum_{i=1}^{n} 9ni^3)[/tex], as n approaches infinity, please follow these steps:
1. Identify the summation notation:
[tex]\sum_{i=1}^{n} 9ni^3[/tex]
2. Calculate the sum using the formula for the sum of cubes:
[tex]\sum_{i=1}^{n} i^3 = (n(n+1)/2)^2[/tex]
3. Substitute the formula into the given expression:
[tex]\lim_{n \to \infty}n(9n(n(n+1)/2)^2)[/tex]
4. Simplify the expression by multiplying n with the sum:
[tex]\lim_{n \to \infty}(9n^2(n(n+1)/2)^2)[/tex]
5. Apply the limit as n approaches infinity.
As n approaches infinity, the term [tex]n^2(n(n+1)/2)^2[/tex] will dominate the expression.
Since this term grows without bounds, the limit of the expression does not exist, or it can be said that the limit is infinity. So, the limit of the given expression as n approaches infinity is infinity.
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Help me guys please i know you are smart
Answer:
B.
Step-by-step explanation:
32, 38, 39, 40, 41, 44, 46, 47
Those are all the ages of the older than 30, but younger than 50.
write the particular solution when k = 0.8. find the time of sale assuming that the goat is sold when its weight reaches 170 pounds. round the answer to the nearest hundredth if necessary.
The time of sale assuming that the goat is sold when its weight reaches 170 pounds is 0.66.
Assuming that the equation relates the weight of a goat to the time elapsed since birth, the general solution can be written as [tex]W(t) = Ce^{(kt)}[/tex], where W(t) is the weight of the goat at time t, C is a constant determined by the initial weight, and k is a constant related to the growth rate of the goat.
To find the particular solution when k = 0.8, we need to know the initial weight of the goat.
Let's assume that the goat weighed 100 pounds at birth, so C = 100.
Therefore, the particular solution is [tex]W(t) = 100e^{(0.8t)}[/tex].
To find the time of sale when the goat weighs 170 pounds, we need to solve for t in the equation W(t) = 170:
[tex]170 = 100e^{(0.8t)}[/tex]
Dividing both sides by 100:
[tex]1.7 = e^{(0.8t)}[/tex]
Taking the natural logarithm of both sides:
ln(1.7) = 0.8t
Solving for t:
t = ln(1.7) / 0.8 ≈ 0.66
Therefore, the goat will be sold when it reaches a weight of 170 pounds after approximately 0.66 units of time (which could be days, weeks, or months depending on the context). Rounded to the nearest hundredth, the time of sale is 0.66.
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a 3.0 kg particle is located on the x-axis at x = −7.0 m and a 5.0 kg particle is on the x axis at x = 3.0 m. what is the center of mass of this two–particle system?
Answer: At x = -0.75
Step-by-step explanation:
Find the center of mass by plugging into the equation.
(3.0 * -7.0 + 5.0 * 3.0) / (3.0 + 5.0) = -0.75
A single die is rolled twice. Find the probability of rolling an odd number the first time and a number greater than 3 the second time. THE Find the probability of rolling an odd number the first time and a number greater than 3 the second time. (Type an integer or a simplified fraction.)
The probability of rolling an odd number the first time and a number greater than 3 the second time is 1/4
Calculating the probabilityThe probability of rolling an odd number on a single die is 3/6 or 1/2, since there are three odd numbers (1, 3, and 5) out of six possible outcomes.
The probability of rolling a number greater than 3 on a single die is 3/6 or 1/2, since there are three numbers greater than 3 (4, 5, and 6) out of six possible outcomes.
To find the probability of both events happening together (rolling an odd number first and a number greater than 3 second), we need to multiply their individual probabilities:
P(odd number first and number > 3 second) = P(odd number first) * P(number > 3 second)
P(odd number first and number > 3 second) = (1/2) * (1/2) = 1/4
Therefore, the probability is 1/4.
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what is the greatest common factor of 400 and 560?
i need an answer asap
80 is the greatest common factor .
What is a factor in math?
A number or algebraic expression that divides another evenly, i.e. without leaving a residue, is referred to in mathematics as a factor. For instance, the precise values of 12 3 = 4 and 12 6 = 2 show that 3 and 6 are factors of 12. 1, 2, 4, and 12 are additional factors of 12.
the prime factorization of 400
400 = 2 × 2 × 2 × 2 × 5 × 5
the prime factorization of 560
560 = 2 × 2 × 2 × 2 × 5 × 7
the GCF, multiply all the prime factors common to both numbers:
Therefore, GCF = 2 × 2 × 2 × 2 × 5
GCF = 80
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Sketch the graphs. 4x+3y=3
For sketching a graph we can give our own x-coordinate points and find the corresponding y-coordinate points.
What are x-coordinates and y-coordinates?In the context of a coordinate system, the x-coordinate (also called the abscissa) is the horizontal distance from the origin to a point on the plane. The y-coordinate (also called the ordinate) is the vertical distance from the origin to the same point on the plane. Together, the x-coordinate and y-coordinate of a point on the plane define the point's position in the coordinate system.
Given equation is 4x+3y=3. For sketching the graph we can give our own x values and find the corresponding y-coordinates. This way we can find the points on the graph. For example, let x = 1, then
4(1) + 3y = 3
3y = 3 - 4 = -1
Hence the point will be (1,-1).
By continuing this process the graph in the image is formed.
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If f(x) = 2x2 – 3x + 5, find f'(o). Use this to find the equation of the tangent line to the parabola y = 2x2 – 3x + 5 at the point (0,5). The equation of this tangent line can be written in the form y = mx + b where m is: and where b is:
The equation of the tangent line is:
y = -3x + 5
To find f'(x), we need to take the derivative of f(x) with respect to x. Given f(x) = 2x^2 - 3x + 5, the derivative f'(x) is:
f'(x) = 4x - 3
Now, we need to find f'(0):
f'(0) = 4(0) - 3 = -3
So the slope (m) of the tangent line at point (0, 5) is -3. Since the tangent line touches the parabola at (0, 5), we can use this point to find the equation of the tangent line:
y = mx + b
Substitute the point (0, 5) and the slope m = -3:
5 = -3(0) + b
5 = b
Thus, the equation of the tangent line is:
y = -3x + 5
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metal gallium is a liquid at room temperature. Its melting point is about 30°C. The freezing point of water is 0°C. How much warmer is the melting point of Gallium than the freezing point of water.
Answer:
Step-by-step explanation:
30 degrees C
3.56 the joint density function of the random variables x and y is f(x, y) = 6x, 0 0.3 | y = 0.5).
It is not possible to find P(y = 0.5) using the given joint density function f(x, y) = 6x, as the value y = 0.5 is outside the defined range for y (0 ≤ y ≤ 0.3).
How to find find P(y = 0.5) joint density function of random variables x and y?Given the joint density function f(x, y) = 6x for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 0.3, you want to find P(y = 0.5). Let's break this down step by step:
1. Understand the problem: We are given the joint density function f(x, y) = 6x with the domain 0 ≤ x ≤ 1 and 0 ≤ y ≤ 0.3. We need to find the probability P(y = 0.5).
2. Identify the issue: Since the joint density function is only defined for y values within the range of 0 ≤ y ≤ 0.3, it is not possible to find P(y = 0.5) using this function, as y = 0.5 is outside the given range.
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Answer this math question for ten points :)
The trigonometric ratios are given as follows:
sin(A) = 4/5.cos(A) = 3/5.tan(A) = 4/3.sin(B) = 3/5.cos(B) = 4/5.tan(B) = 3/4.What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.5 is the hypotenuse length, while for angle A, we have that the sides are given as follows:
Opposite side of 4.Adjacent side of 3.Hence the ratios are given as follows:
sin(A) = 4/5.cos(A) = 3/5.tan(A) = 4/3.For angle B, we have that 4 is now the adjacent side, while 3 is the opposite side, hence:
sin(B) = 3/5.cos(B) = 4/5.tan(B) = 3/4.More can be learned about trigonometric ratios at brainly.com/question/24349828
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According to Interactive Illustration 1: Porter's Forces Framework in the Core Reading, the impact of a reduced threat of entry is that: a. profitability increases because willingness to pay increases, prices increase, and costs decrease 0. b. profitability increases because willingness to pay decreases, prices increase, and costs decrease. c. profitability decreases because willingness to pay decreases, prices decrease, and costs increase d. profitability decreases because willingness to pay decreases, prices decrease, and costs decrease.
The correct answer is: c. profitability decreases because the willingness to pay decreases, prices decrease, and costs increase. Porter's Five Forces Framework is a strategic management tool developed by Harvard Business School professor Michael E. Porter. The framework is used to analyze the competitive environment of an industry and to identify the factors that influence the profitability of a company within that industry.
According to Porter's Five Forces Framework, a reduced threat of entry means that the market is less competitive, and therefore, the existing firms have more market power. This can lead to a decrease in the willingness to pay for customers, as they have fewer choices and are less likely to switch to competitors. As a result, firms may need to lower their prices to maintain demand, which can decrease profitability. Additionally, because there is less pressure to compete, firms may not invest in cost-cutting measures, leading to increased costs. Therefore, profitability decreases due to lower willingness to pay, lower prices, and higher costs.
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Ahmed is modelling the area covered by a moss using the equation A = t² + 5.8t +
9.41 where A is the area covered in square centimeters, negative values of t represent
a number of days after noon on July 1, 2014. Which of the following equivalent
expressions contain the number of days before noon on July 1, 2014, as a constant or
coefficient, when the moss started with its smallest area of 1 square centimeter?
a.(t − (−2.9))² + 1
b. (t−(−1))² +3.8t +8.41
c.(t-(-3.9)) (t- (-1.9)) + 2
d. t(t− (−5.8)) + 9.41
Answer:
a. (t - (-2.9))² + 1
Step-by-step explanation:
We are told that Ahmed is modelling the area covered by a moss using the equation A = t² + 5.8t + 9.41.
The given equation is a quadratic equation with a positive leading coefficient. Therefore, the graph is a parabola that opens upwards. This means that the vertex is its lowest point.
The vertex form of a quadratic equation is:
[tex]\boxed{y = (x - h)^2 + k}[/tex]
where (h, k) is the vertex.
As the vertex is the lowest point, this means that the y-value of the vertex is when the area of moss is at its lowest. We are told that the smallest area of the moss is 1 square centimeter, so k = 1.
The x-value of the vertex can be found by using the formula:
[tex]h=-\dfrac{b}{2a}[/tex]
for a quadratic in the form ax² + bx + c.
For the given quadratic, a = 1, b = 5.8 and c = 9.41.
Substitute the values of a and b into the formula to find the x-value of the vertex:
[tex]\implies h=-\dfrac{5.8}{2(1)}=-2.9[/tex]
Finally, substitute h = -2.9 and k = 1 into the vertex form:
[tex]A=(t-(-2.9))^2+1[/tex]
Therefore, the equivalent expression that contains the number of days before noon on July 1, 2014, as a constant or coefficient, when the moss started with its smallest area of 1 square centimeter is:
(t - (-2.9))² + 1One side of a triangle is 84cm the other two sides are in the ratio 3:8 If the perimeter is 282cm find the the longest and shortest side
Use for Problems 6-9: A large supermarket stocks both national brands of coffee and its own house brand. Consider a single randomly selected customer purchasing coffee and let success = the customer purchases a national brand. Assume that p = 0.75 and that customers make coffee purchase decisions independently of one another. Use R to calculate the probabilities. 6. Let X = number of coffee purchasers who select a national brand from the 10 randomly selected customers purchasing coffee. a. Which distribution should we use? b. Find the probability exactly 4 of the 10 will purchase a national brand from the 10 randomly selected customers purchasing coffee. (answer to 4 decimal places) Insert your code here: Answer: C. Find the probability that at most 7 will purchase a national brand from the 10 randomly selected customers purchasing coffee. (answer to 4 decimal places)
A. We should use the binomial distribution since we are interested in the number of successes (customers who purchase a national brand) out of a fixed number of trials (10 customers).
b. To find the probability exactly 4 of the 10 customers will purchase a national brand, we can use the dbinom function in R:
dbinom(4, 10, 0.75)
The answer is 0.2503 (to 4 decimal places).
c. To find the probability that at most 7 customers will purchase a national brand, we can use the pbinom function in R:
pbinom(7, 10, 0.75)
The answer is 0.9831 (to 4 decimal places).
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Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
The coordinate axes and the line x + y = 4
The area of the region bounded by the line x + y = 4 and the coordinate axes is calculated to be 8/3 square units.
The region bounded by the line x + y = 4 and the coordinate axes is a right triangle with vertices at (0,0), (4,0), and (0,4). To express the area of this region as an iterated double integral, we can integrate over the rectangle R = [0,4] × [0,4] and subtract the integral over the triangle T = {(x,y) : x + y ≤ 4}.
Thus, the area of the region is given by the double integral:
A = ∬R dA - ∬T dA
Since dA = dxdy, we can evaluate this as:
A = ∫0⁴ ∫0⁴ dxdy - ∫0⁴ ∫0⁴-x+y dxdy
Simplifying this, we get:
A = ∫0⁴ ∫0⁴-x dydx
Evaluating the inner integral first, we get:
A = ∫0⁴ (-x)(4-x) dx
Integrating this, we obtain:
A = ∫0⁴ (-4x + x²) dx = [-2x^2 + (1/3)x³]0⁴ = 8/3
Therefore, the area of the region bounded by the line x + y = 4 and the coordinate axes is 8/3 square units.
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RS≅ST, m∠RST=7x - 54, m∠STU = 8x
Answer:
Step-by-step explanation:
find the scale factor of the dilation
The value of a certain investment over time is given in the table below. Answer the
questions below to determine what kind of function would best fit the data, linear or
exponential.
Number of
Years Since
Investment
Made, x
Value of
Investment
(8), f(x)
11.486.36
9,181.76
values change
function is approximately
3
6,890.96
4.581.76
function would best fit the data because as x increases, the y
The
of this
The slope of this function is approximately 4518
How to solveA linear function would best fit the data because as x increases, the y values change values by 4518.
y = mx + c
Linear equation with two variables, when graphed on the cartesian plane with axes of those variables, give a straight line.
the linear function would best fit the data because as x increases, the y values change values by 4518.
The slope of this function is approximately 4518
Slope = change in y values / change in x values
=( 27520.99-23002.99)/(2-1)
= 4518/1
= 4518
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3,200 divided by 1000 in long division
Answer:
Step-by-step explanation:
1000 | 3200
-3000
----
200
Therefore, 3,200 divided by 1000 is equal to 3.2.
15% of the students in the library are girls. If there are 60 total students in the library, how many are girls?
Answer:
9 girls
Step-by-step explanation:
15% x 60 = 9
Answer: 15% of 60 is 9.
Step-by-step explanation:
15% of 60 can be written as 15% × 60
= 15/100 × 60
= 9
Let X be a discrete random variable with probability mass function given byP(x)={c/4 x=0{c/4 x=1{c x=2{0 otherwiseFind the value of that makes p a valid probability mass function.
In order for a probability mass function (PMF) to be valid, it must satisfy two conditions:
Condition 1- Non-negativity:
c/4, c/4 and c are all non-negative values.
Condition 2- Sum of PMF equals 1:
The sum of the PMF over all possible values of x must be equal to 1.
P(0) + P(1) + P(2) + P(x) for all other x
= c/4 + c/4 + c + 0 (since P(x) = 0 for all other x)
= (c + c + 4c)/4
= 6c/4
In order for this sum to be equal to 1, we must have:
6c/4 = 1
Multiplying both sides by 4/6 to solve for c:
c = 4/6
c = 2/3
So, the value of c that makes P(x) a valid probability mass function is c = 2/3.
To find the value of c that makes P a valid probability mass function, we need to ensure that the sum of all probabilities equals 1.
We can do this by summing the probabilities for all possible values of X:
P(0) + P(1) + P(2) = c/4 + c/4 + c = 1
Simplifying the equation:
c/2 + c = 1
3c/2 = 1
c = 2/3
Therefore, the value of c that makes P a valid probability mass function is 2/3.
To find the value of c that makes P(x) a valid probability mass function, we need to ensure that the sum of probabilities for all possible values of x is equal to 1. Given the probability mass function:
P(x) = {c/4, x=0;
c/4, x=1;
c, x=2;
0, otherwise}
The sum of probabilities for all x values should be:
P(x=0) + P(x=1) + P(x=2) = 1
Substituting the given values:
(c/4) + (c/4) + c = 1
Now, we solve for c:
c/4 + c/4 + c = 1
(2c/4) + c = 1
(1/2)c + c = 1
(3/2)c = 1
To find the value of c:
c = 1 / (3/2)
c = 1 * (2/3)
c = 2/3
So the value of c that makes P(x) a valid probability mass function is 2/3.
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What do we get on multiplying both sides of −1 < −x/4 by −4 ?
The inequality is 4<-x.
What is inequality?
The term "inequality" is used in mathematics to describe a relationship between two expressions or values that is not equal to one another. Inequality results from a lack of balance. When two quantities are equal, we use the symbol '=', and when they are not equal, we use the symbol. If two values are not equal, the first value can be greater than (>) or less than (), or greater than equal to () or less than equal to ().
Here the given inequality is,
=> -1 < [tex]\frac{-x}{4}[/tex]
Now multiply by -4 on both sides then,
=> [tex]-1\times-4 < \frac{-x}{-4}\times-4[/tex]
=> 4 < -x
Hence the inequality is 4<-x.
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help!!!
see pic below⬇
Answer:
10:00 am > 0 (initial time) > 24 gallons10:30 > 30 minutes > 40 gallons 11:00 am > 1 hour (from initial time) > 56 gallonsStep-by-step explanation: every 30 minutes we add 16 to the level
Answer:for 10:am youd put "started" or "0" for time and "24 gallons of water" at 10:30am put "30 minutes" for time because from 10 to 10:30 is 30 minutes. For amount of water put 40 gallons. For the last one put the tine as "1:00pm". Put the time spent as "2 hours and 30 min" and gallons of water put 120
please help i’ll mark brainliest
The type of quadrilaterals, based on the description of the diagonals are;
a) Isosceles trapezoid
b) Square or rhombus
c) Kite or parallelogram
What is a quadrilateral?A quadrilateral is a four sided polygon.
The properties of the quadrilateral are;
a) The diagonals are congruent but are not perpendicular
b) The diagonals are congruent and perpendicular bisectors
c) One diagonal bisects an angle and the other diagonal
The type of quadrilaterals are;
a) Whereby the diagonals are congruent and the diagonals are not perpendicular, indicates that a possible quadrilateral is an isosceles trapezoid
b) The quadrilaterals with congruent and perpendicular diagonals indicates that the possible quadrilaterals are a square or a rhombus
c) A quadrilateral that bisects an angle and one diagonal indicates that the quadrilateral is a parallelogram or kite
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Problems 11 through 23 and Case Problem 2 require the use of data mining software. If using R/Rattle to solve these problems, refer to Appendix: R/Rattle Settings to Solve Chapter 5 Problems. If using JMP Pro to solve these problems, refer to Appendix: JMP Pro Settings to Solve Chapter 5 Problems.
Association Rules of Browser Histories. Cookie Monster Inc. is a company that specializes in the development of software that tracks web browsing history of individuals. Cookie Monster Inc. is interested in analyzing its data to gain insight on the online behavior of individuals. A sample of browser histories is provided in the files CookieMonsterBinary and CookieMonsterStacked that indicate which websites were visited by which customers. Use a minimum support of 4% of the transactions (800 of the 20,000 total transactions) and a minimum confidence of 50% to generate a list of association rules.
a. Based on the top 14 rules, which three web sites appear in the association rules with the largest lift ratio?
b. Identify the association rule with the largest lift ratio that also has Pinterest as the antecedent. What is the consequent web site in this rule?
c. Interpret the confidence of the rule from part (b). While the antecedent and consequent are not necessarily chronological, what does this rule suggest?
d. Identify the association rule with the largest lift ratio that also has TheEveryGirl as the antecedent. What is the consequent web site in this rule?
e. Interpret the lift ratio of the rule from part (d).
The lift ratio indicates a strong association between the two websites.
Association rule mining is a data mining technique used to find patterns in datasets. It is particularly useful for finding relationships between variables in large datasets. In this case, the dataset consists of browser histories, and we are interested in finding association rules between the websites that were visited.
The two main measures used in association rule mining are support and confidence. Support is the proportion of transactions that contain both the antecedent and the consequent of a rule, while confidence is the proportion of transactions containing the antecedent that also contain the consequent. Lift ratio is another measure used to evaluate the strength of an association rule. It is the ratio of the observed support to the expected support, assuming the antecedent and the consequent are independent.
To generate a list of association rules, we can use an algorithm like Apriori. The Apriori algorithm starts by finding frequent itemsets, or sets of items that occur together in a sufficient number of transactions. We can then use these frequent itemsets to generate association rules by dividing them into antecedents and consequents and calculating the support and confidence of each rule.
Based on the top 14 rules, we can identify the three web sites that appear in the association rules with the largest lift ratio. Lift ratio is a measure of how much more often the antecedent and consequent appear together than would be expected if they were independent. Therefore, high lift ratios indicate strong associations between the antecedent and consequent. We can sort the rules by lift ratio and identify the three web sites that appear in the rules with the highest lift ratios.
To identify the association rule with the largest lift ratio that also has a specific antecedent, we can filter the rules by the antecedent and sort them by lift ratio. We can then examine the consequent of the rule with the highest lift ratio.
The confidence of a rule indicates how often the consequent is observed in transactions containing the antecedent. In this case, the rule suggests that there is a high probability that customers who visit Pinterest also visit the consequent website. While the antecedent and consequent are not necessarily chronological, the rule suggests that there may be a relationship between the two websites that prompts customers to visit them together.
The lift ratio of a rule indicates the strength of the association between the antecedent and consequent. In this case, the rule suggests that customers who visit TheEveryGirl are much more likely to also visit the consequent website than would be expected if the two websites were independent. Therefore, the lift ratio indicates a strong association between the two websites.
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Find the volume of the solid whose base is the region bounded between the curve y=x² and the x-axis from x=0 to x=2 and whose cross sections taken perpendicular to the x-axis are squares.
Minimizing Surface Area:
(i) Of all boxes with a square base and a fixed volume V, which one has the minimum surface area As? (Give its dimensions in terms of V.)
(ii) Let V = 1000 meters cubed and give the dimensions using your solution in part (i).
(iii) Sketch the surface area function A that was minimized in part (i). Use a reasonable domain. Label axes appropriately, including units.
To minimize the surface area of a box with a square base and a fixed volume V, its dimensions should be x = y = z = (V¹/³). For V = 1000 meters cubed, the dimensions are x = y = z = 10 meters.
To minimize the surface area, we can use the formula for the surface area of a box with a square base: A = 2x² + 4xy, where x = y (square base) and z = V/x². Differentiating A with respect to x and setting the derivative equal to zero, we find the critical points.
A' = 4x - 4V/x³. Setting A' = 0, we get 4x = 4V/x³, and x⁴ = V, so x = (V¹/³). Since x = y, the dimensions are x = y = z = (V¹/³).
For part (ii), let V = 1000 meters cubed. Then, the dimensions are x = y = z = (1000)¹/³ = 10 meters.
For part (iii), sketch the surface area function A(x) = 2x²+ 4x(V/x²) with a reasonable domain, such as [1, 20] meters for x-axis and [300, 2500] meters squared for the y-axis. Label axes appropriately, including units.
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On Your Own
1) A sphere has a radius of 7 cm. What is the volume?
round your answer to the
nearest cubic centimeter.
Like example 1
4
2) A sphere has a volume of 72π in³. What is the radius?
round to the nearest tenth
of an inch.
Like example 3
hope this helps you
find a formula for the general term an (not the partial sum) of the infinite series (starting with a1). 13 +19 + 127 +181 ⋯
The formula for the general term of the given series is:
[tex]a_n = 13 + (n-1)6^{(n-1)[/tex], n >= 1
This formula gives us the nth term of the series by adding 6 raised to the [tex](n-1)^{th[/tex] power to the first term 13.
How to find a formula for the general term of the series?To find a formula for the general term of the infinite series, we need to look for a pattern in the given terms.
Notice that if we add 6 to the first term 13, we get the second term 19. Similarly, if we add 6 to the second term 19, we get the third term 127.
And if we add 6 to the third term 127, we get the fourth term 181. So the difference between consecutive terms is not constant, but it seems to be increasing by a factor of 6 each time.
Let's check this by finding the difference between consecutive terms:
19 - 13 = 6
127 - 19 = 108
181 - 127 = 54
Indeed, the differences are 6, 6 times 18, and 6 times 9, which confirms that the pattern we observed holds.
So we can write the general term as follows:
[tex]a_n = 13 + (n-1)6^{(n-1)[/tex], n >= 1
This formula gives us the nth term of the series by adding 6 raised to the [tex](n-1)^{th[/tex] power to the first term 13.
Learn more about formula for the general term of an infinite series.
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5. Tony measured a TV that is approximately 14 inches tall by 24 inches wide. Since the size of TVs are named by the measure of the diagonal, what is the BEST diagonal measure, in inches, for this TV?
A. 772 inches
B. 28 inches
C. 76 inches
D. 9 inches
Answer:
Using this formula, we can calculate the diagonal measure of the TV as follows: (this is the Phytagoras theorem)
diagonal^2 = height^2 + width^2
diagonal^2 = 14^2 + 24^2
diagonal^2 = 196 + 576
diagonal^2 = 772
diagonal ≈ 27.8 inches
Therefore, this TV's best diagonal measure, in inches, is option B, 28 inches (rounded to the nearest inch).