Answer:
301.4 in^3
Step-by-step explanation:
Divide 12 by 2 to get the radius
12/2=6
Square 6 and multiply by 3.14 according to the area of a circle formula πr^2.
6^2=36 36*3.14=113.04
We now have the area of the base.
Multiply it by the height and divide by three to follow the formula for the volume of a cone, 1/3bh.
113.04*8=904.32. 904.32/3= 301.44, or 301.4 after rounding.
A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.
10, 11, 35, 39, 40, 42, 42, 45, 49, 49, 51, 51, 52, 53, 53, 54, 56, 59
A graph titled Donations to Charity in Dollars. The x-axis is labeled 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 10 to 19, up to 2 above 30 to 39, up to 6 above 40 to 49, and up to 8 above 50 to 59. There is no shaded bar above 20 to 29.
Which measure of variability should the charity use to accurately represent the data? Explain your answer.
The range of 13 is the most accurate to use, since the data is skewed.
The IQR of 49 is the most accurate to use to show that they need more money.
The range of 49 is the most accurate to use to show that they have plenty of money.
The IQR of 13 is the most accurate to use, since the data is skewed.
The most appropriate measure of variability to use in this case is the IQR (Interquartile Range) of 13.
How to determine which measure of variability should the charity use to accurately represent the dataThe reason for this is that the data appears to be skewed towards the higher values, with more donations in the higher range (from 40 to 59 dollars) and fewer in the lower range (from 10 to 39 dollars).
Using the range (which is the difference between the highest and lowest values) would not accurately represent the variability of the data, as it would be heavily influenced by the outliers. On the other hand, the IQR, which is the range of the middle 50% of the data, is a more robust measure of variability that is not influenced by outliers. Therefore, the IQR of 13 is the most appropriate measure of variability to use in this case.
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If a cylinder has a height of 5 inches and a radius of 3 inches, which equation can be used to find V, the volume of the cylinder in cubic inches?
The equation to find the volume of the cylinder is: V = 45π cubic inches.
What is cubic inches?The volume of things or containers is often measured in cubic inches in the United States. It is the amount of area that a cube with one inch sides takes up. A cubic inch is about equal to 16.387 millilitres or 0.016387064 litres. The displacement of an engine—a measurement of the total amount of air and fuel the engine can compress into its cylinders—is frequently discussed in terms of cubic inches.
According to given information:The formula to find the volume of a cylinder is:
V = πr²h
Where V is the volume, r is the radius, and h is the height.
Substituting the values given in the problem, we get:
V = π(3²)(5)
V = π(9)(5)
V = 45π
Therefore, the equation to find the volume of the cylinder is:
V = 45π cubic inches.
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Find the distance between the two points (3,−2) and (2,6) .
Simplify your answer, and write the exact answer in simplest radical form for an irrational answer. For example, 2–√= sqrt(2) .
Answer:
[tex] \sqrt{( {3 - 2)}^{2} + {( - 2 - 6)}^{2} } [/tex]
[tex] \sqrt{ {1}^{2} + {( - 8)}^{2} } [/tex]
[tex] \sqrt{1 + 64} = \sqrt{65} [/tex]
Confusing Calculus Parametric Equations Question
Rounding to the nearest meter, we get that the bullet will hit the ground 18603 meters away from the gun.
what is acceleration?The rate at which an object's velocity varies with respect to time is known as its acceleration. It is a vector quantity whose definition is the product of the velocity change and the time period during which the change takes place. Depending on the direction and size of the velocity change, acceleration may be positive, negative, or zero. Acceleration in physics is frequently represented by the letter "a," and its SI unit is metres per second squared (m/s2).
To find when the bullet will hit the ground, we need to find the time at which y = 0.
y = vo sin(a) t - (1/2)g[tex]t^2[/tex], where g is the acceleration due to gravity (9.8 m/s^2).
Setting y = 0, we get:
0 = vo sin(a) t - (1/2)g[tex]t^2[/tex]
Solving for t using the quadratic formula, we get:
Since we want the time when the bullet hits the ground, we take the positive root:
t = (vo sin(a) + √((vo sin(a))^2 + 2gy)) / g
Plugging in the values given, we get:
t = (550 m/s * sin(a) + √((550 m/s * sin(a))^2 + 2 * 9.8 m/s^2 * 0)) / 9.8 m/s^2
Simplifying, we get:
t = (550 m/s * sin(a)) / 9.8 m/s^2
t = 35.27 seconds (rounded to two decimal places)
To find how far from the gun the bullet will hit the ground, we need to find the value of x at this time:
x = vo cos(a) t
Plugging in the values given, we get:
x = 550 m/s * cos(a) * 35.27 s
x = 18602.57 meters
Rounding to the nearest meter, we get that the bullet will hit the ground 18603 meters away from the gun.
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please help
What is the distance to the earth’s horizon from point P?
Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.
x =
mi
The measure of distance x, that is, the distance between a point P and the point of horizon, is equal to 284.372 miles.
How to find the distance to the earth horizon from a given point
In this problem we must determine the distance between a point P located about earth's circumference and the point of horizon, located on earth's circumference. Since the line between these two points is tangent to earth, then, distance x can be found by Pythagorean theorem:
x = √[(3959 mi + 10.2 mi)² - (3959 mi)²]
x = 284.372 mi
The distance x is equal to 284.372 miles.
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You are trying to determine if a router you’re thinking of purchasing will give off a strong enough wifi signal to cover the area in your house. The router you have selected will cover an area of 850 ft^2.
According to the information, the router cannot cover the kitchen area.
Does the router cover the kitchen area?To calculate if the router covers the kitchen area, we must identify the area that the router can cover. In this case, the router covers an area of 850 square feet. However, we must take into account that this area is counted around the router. So, we have to calculate the square root of 850 ft^2.
[tex]\sqrt{850}[/tex] = 29.129.1 / 2 = 14.5From the above, we can infer that the router has signal coverage of 14 meters around it, so it would only cover the living room area.
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Factor the expression. x^2-14x-32
Answer:
[tex](x-2)(x+16)[/tex]
Step-by-step explanation:
Use graphing calculator and find the x-intercepts.
Please help me understand this
Answer:
To find the value of y, we can use the slope formula:
slope = (y2 - y1) / (x2 - x1)
We know that the slope is 5, and the two points on the line are (5,y) and (4,1). We can substitute these values into the slope formula to get:
5 = (y - 1) / (5 - 4)
Simplifying this equation gives:
5 = y - 1
Adding 1 to both sides gives:
y = 6
Therefore, the value of y is 6.
Step-by-step explanation:
For each graph below, state whether it represents a function. I don’t know if I got them right
Answer:
Step-by-step explanation:
Use the vertical line test. If you can draw a vertical line that cuts more than one point then the graph is not a function.
The only one that is incorrect is graph 4. It is a function (no vertical line will cut the graph more than once).
El perímetro de un campo rectangular es 300 m . Si la longitud del campo es 88 , ¿cuál es su anchura?
Based on the above, the width of the field is 62 meters.
What is the width?Based on the question, Let's say that the width of the field is denoted with 'w'.
Note that the formula for the perimeter (P) of a rectangle is:
P = 2(l + w)
where"
l = length
w = width.
Fixing the values into the equation, it will be:
300 = 2(88 + w)
So, Divide both sides by 2:
150 = 88 + w
Subtract 88 from both sides:
w = 150 - 88
w = 62
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Se text below
The perimeter of a rectangular field is 300 m. If the length of the field is 88 , what is its width?
Suppose that the length of the three sides of a right triangle are consecutive whole numbers. What is the length of the longest leg?
The length of the longest leg in the right triangle is 5 units
What is the length of the longest leg?Let x be the length of the shortest leg. Then the other leg has length x + 1, and the hypotenuse has length x + 2.
By the Pythagorean Theorem, we have:
x^2 + (x + 1)^2 = (x + 2)^2
Set x = 3
So, we have
3^2 + (3 + 1)^2 = (3 + 2)^2
Evaluate
25 = 25
The above equation is trie
So, we have
Longest = x + 2
Longest = 3 + 2
Evaluate
Longest = 5
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Which expression represents the volume of a shipping box? s(2^3) (8 + 2)3 (8x2)3 8+23
As a result, the right response is s(23), which can be written as [tex]8s^3[/tex] as to calculate a shipping box's volume is based on the box's dimensions.
what is volume ?A tri object's volume is a measurement of how much space it takes up. The usual measurement tool is a cubic unit, such as a cubic metre (m3) or cubic foot (ft3). The formula used to determine a solid item's volume depends on the shape of the object. For instance, the volume of a cube is determined by multiplying the length of a single of its sides by six, whereas the capacity of a cylinder is determined by dividing the area of its base by the diameter. Engineering, mathematics, and architecture are just a few of the disciplines where the idea of volume is crucial.
given
The formula used to calculate a shipping box's volume is based on the box's dimensions.
The box's volume can be expressed as follows if its side length is s:
[tex]V = s^3[/tex]
As a result, the right response is s(23), which can be written as [tex]8s^3[/tex] as to calculate a shipping box's volume is based on the box's dimensions.
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The Celluloid Collar Corporation has $120,000 in tax loss carryforwards. The Bowstring Shirt Company, a firm in the 21% tax bracket, would be willing to pay (on a non-discounted basis) the sum of _________ for Celluloid Collar Corporation’s carryforward alone.
Group of answer choices
$18,000
$25,200
$44,100
$20,000
The Bowstring Shirt Company, a firm in the 21% tax bracket, would be willing to pay (on a non-discounted basis) the sum of $25,200 for Celluloid Collar Corporation’s carryforward alone.
What is the non-discounted sum?The value of Celluloid Collar Corporation's tax loss when it is said to be carryforward to Bowstring Shirt Company will be be calculated using the steps below
Value = Tax Loss Carryforward x Tax Rate
Value = $120,000 x 21%
= $25,200
Therefore, Bowstring Shirt Company have to pay $25,200 for Celluloid Collar Corporation's tax loss to be carryforward.
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11. Reason Inductively Complete the statement: If 8 = 2, then
m-12=. Explain.
The complete statement is, if 8 = m/4, then m - 12 = 20.
Describe Algebra?Algebra is a branch of mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols. It involves solving equations and inequalities, manipulating expressions, and working with mathematical formulas to describe and analyze relationships between variables. Algebraic equations and expressions are used to represent and solve problems in a wide range of fields, including science, engineering, economics, and finance. Algebra is an essential tool for problem-solving and critical thinking, and it provides the foundation for more advanced mathematical concepts such as calculus, statistics, and linear algebra.
The main goal of algebra is to find unknown values or variables, typically represented by letters such as x, y, or z. Algebraic equations involve a variety of mathematical operations, including addition, subtraction, multiplication, and division. These equations may also include exponents, logarithms, and other advanced mathematical concepts.
To solve the problem, we can use algebraic manipulation.
Given 8 = m/4, we can solve for m by multiplying both sides by 4:
8 × 4 = m
32 = m
Now we can substitute m = 32 into the expression m - 12 to find:
m - 12 = 32 - 12 = 20
Therefore, if 8 = m/4, then m - 12 = 20.
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The complete question is:
Complete the statement: if 8= m/4, then m-12=
You’re planning to buy a house and you want to have $35,000 to put as a down payment. If you have $20,000 today and you’re able to receive 12% interest rate for investing the money, how long will it take you to have enough money for the down payment saved?
It will take approximately 6.24 years to have enough money for the down payment saved.
How to calculate how long will it take you to have enough money for the down payment savedWe can use the formula for compound interest to solve this problem:
A = P(1 + r/n)^(nt)
where:
A = the amount of money we want to have (in this case, $35,000)
P = the initial amount we have (in this case, $20,000)
r = the annual interest rate (in this case, 12%)
n = the number of times interest is compounded per year (let's assume it's compounded monthly, so n = 12)
t = the number of years we're investing for (this is what we want to find)
Substituting the values we know:
$35,000 = $20,000(1 + 0.12/12)^(12t)
Simplifying:
1.75 = (1 + 0.01)^(12t)
Taking the natural logarithm of both sides:
ln(1.75) = 12t ln(1.01)
Dividing both sides by 12 ln(1.01):
t = ln(1.75) / (12 ln(1.01))
Using a calculator:
t ≈ 6.24 years
Therefore, it will take approximately 6.24 years to have enough money for the down payment saved.
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I need help with this econ question. Thank you.
Answer:
c + 3.6 + 3.4 + 4.1 + 2.3 = 18.6
c + 13.4 = 18.6
c = 5.2
$5.2 trillion was spent on consumption.
If $4,500 is invested at an interest rate of 5.25% per year, compounded continuously, find the value of the investment after the given number of years. (Round your answers to the nearest cent.)
(a) 4 years
(b) 8 years
(c) 12 years
a) The value of the investment after 4 years is $5,544.90.
b) The value of the investment after 8 years is $6,849.42.
c) The value of the investment after 12 years is $8,453.35.
The formula for calculating the value of an investment that is compounded continuously is given by:
A = [tex]Pe^{(rt)[/tex]
where A is the final value of the investment, P is the initial investment amount, e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate expressed as a decimal, and t is the time period in years.
(a) To find the value of the investment after 4 years, we can substitute the given values in the formula as follows:
A = 4500[tex]e^{(0.05254)[/tex]
A = 4500[tex]e^{0.21[/tex]
A = 45001.2322
A = $5,544.90
(b) To find the value of the investment after 8 years, we can substitute the given values in the formula as follows:
A = 4500[tex]e^{(0.05258)[/tex]
A = 4500[tex]e^{0.42[/tex]
A = 45001.5221
A = $6,849.42
(c) To find the value of the investment after 12 years, we can substitute the given values in the formula as follows:
A = 4500[tex]e^{(0.052512)[/tex]
A = 4500[tex]e^{0.63[/tex]
A = 45001.8785
A = $8,453.35
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Suppose that the function f is defined as follows.
Answer:
Step-by-step explanation:
Please help me!
Sloan keeps quarters and dimes in a jar, he just counted and has $23.45. If Sloane has 52 dimes in the jar, how many quarters does he have?
Answer in standard form, NEED EQUATION
x =
y =
Sloan has 73 quarters and 52 dimes kept in the jar
What is an equation?An equation is an expression that shows how numbers and variables using mathematical operators.
Let x represent the number of quarters in the jar.
1 dime = $0.10, and 1 quarter = $0.25
Hence:
He counted $23.45, of which he had 52 dimes, hence:
0.10(52) + 0.25x = 23.45
5.2 + 0.25x = 23.45
x = 73
Sloan has 73 quarters
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Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
The two equations that have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p include the following:
B. 2.3p – 10.1 = 6.49p – 4
C. 230p – 1010 = 650p – 400 – p
How to determine the required equations?From the information provided, we have the following equation:
2.3p – 10.1 = 6.5p – 4 – 0.01p
Next, we would simplify the given equation by collecting like terms as follows;
2.3p - 10.1 = 6.5p - 0.01p - 4 = 6.49p - 4
2.3p - 10.1 = 6.49p - 4
What is the multiplication property of equality?In Mathematics, the multiplication property of equality states that both sides of an equation will remain the same and equal, when both sides of the equations are multiplied by the same number.
By multiplying both sides of the given equation by 100, we have the following;
100 × (2.3p - 10.1) = 100 × (6.5p - 4 - 0.01p)
230p - 1010 = 650p - 400 - p
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Leslie invests in a bank account that earns interest compounded quarterly. The expression 500(1.026)4t can be used to find the account balance in t years.
Part A
What was Leslie’s initial investment?
$
Part B
What is the quarterly interest rate the account earns?
Leslie's initial investment was $500 and the quarterly interest rate the account earns is 10.4%.
To find the initial investment and quarterly interest rate we can use the following compound formula of
A = P [tex](1 + \frac{r}{n} )^{nt}[/tex]
Given expression =[tex]500(1.026)^{4t}[/tex]
principal P = $500
rate of interest r = 1.026
A) Leslie's initial investement A = [tex]500 (1.026) ^{0}[/tex] = 500 [ initially time t = 0]
B) for finding quarterly interest,
given expression =[tex]500(1.026)^{4t}[/tex]
=[tex]500(1 + 0.026)^{4t}[/tex]
= [tex]500(1 + 0.026 * 4/4)^{4t}[/tex]
= [tex]500(1 + \frac{0.104}{4} )^{4t}[/tex]
From the above expression, we can conclude that the quarterly interest is 0.104 which is 10.4%
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please find the SA and can you please explain?
Please hurry if you can.
Answer:
Rounding to the nearest hundredth, the surface area of the regular hexagonal pyramid is approximately 663.2 square centimeters.
Step by Step Explanation:
The surface area of a regular hexagonal pyramid can be calculated using the formula:
Surface area = (1/2) x Perimeter of base x Slant height + Base area
To find the perimeter of the hexagonal base, we can use the formula:
Perimeter of hexagon = 6 x Length of one side
Since the base edge length is 8 centimeters, the length of one side of the hexagon is also 8 centimeters. Therefore, the perimeter of the hexagonal base is:
Perimeter of base = 6 x Length of one side = 6 x 8 = 48 centimeters
To find the base area of the pyramid, we can use the formula:
Base area = (3/2) x (Square root of 3) x (Length of one side)^2
Substituting the values, we get:
Base area = (3/2) x (Square root of 3) x (8)^2 = 96(sqrt(3)) square centimeters
Now we can substitute the values into the surface area formula:
Surface area = (1/2) x Perimeter of base x Slant height + Base area
Surface area = (1/2) x 48 x 15 + 96(sqrt(3))
Surface area = 360 + 96(sqrt(3)) square centimeters
If the function y=e−2x is vertically compressed by a factor of 3, reflected across the y-axis, and then shifted down 2 units, what is the resulting function? Write your answer in the form y=ceax+b
The resulting function is given as follows:
[tex]y = \frac{e^{2x}}{3} - 2[/tex]
How to obtain the resulting function?The parent function in the context of this problem is given as follows:
[tex]y = e^{-2x}[/tex]
When the function is vertically compressed by a factor of 3, we have that it is divided by three on the range, hence:
[tex]y = \frac{e^{-2x}}{3}[/tex]
When the function is reflected across the x-axis, we multiply by -1 on the domain, hence:
[tex]y = \frac{e^{2x}}{3}[/tex], as -2x x -1 = 2x.
When it is shifted down two units, we subtract by two on the range, hence:
[tex]y = \frac{e^{2x}}{3} - 2[/tex]
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How do I help my student understand this?
Cell 1 is in the "7th grade" row and "Pets" column. Have your student circle these labels or highlight them. Then it should be clear why choice E is the final answer.
Another way to rephrase it: the 35 people in cell 1 represent the number of 7th graders with pets.
Match the situation with its probability.
The given probabilities are as follows:
Spinner landing on at least one A = 7/16Spinner landing on C and D in any order = 1/8Spinner landing on two Bs = 1/16Spinner landing on C on the second spin = 1/4What are the probabilities?The probabilities are determined using the formula below:
Probability = number of expected outcome / number of total outcomesSpinner landing on at least one A:
number of at least one A = 7
Spinner landing on at least one A = 7/16
Spinner landing on C and D in any order:
number of landing on C and D in any order = 2
Spinner landing on C and D in any order = 2/16
Spinner landing on C and D in any order = 1/8
Spinner landing on two Bs:
number of landing on two Bs = 1
Spinner landing on two Bs = 1/16
Spinner landing on C on the second spin:
Number of landing on C on the second spin = 4
Spinner landing on C on the second spin = 1/4
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PLS HELP THIS IS DUE TOMORROW
Answer: y = -|x + 1| + 1
Step-by-step explanation:
Since this is a "V" shaped graph, we know it uses the absolute value function. This is the parent function:
y = |x|
This represents the transformations:
➜ a is amplitude
➜ h is horizontal shift
➜ k is vertical shift
f(x) = a | x - h | + k
Next, we see it is shifted one unit upwards.
y = |x| + 1
Then, we see it is also shifted one unit left.
➜ Note that this shift is -h units, so we will use positive for moving left.
y = |x + 1| + 1
Lastly, we see this graph is flipped and has a negative slope, or amplitude.
y = -|x + 1| + 1
Wendy is customizing shirts for the debate team. She can choose from 8 styles, 7 colors, and 7 collars. How many different shirts is it possible for Wendy to customize?
Answer:
392 conbinations
Step-by-step explanation:
multiply all 3 numbers to find conbinations.
8x7x7
=56x7(or 49x8)
=392
its prob wrong cuz i guessed so yeah
Simplify. Simplify. Express your answer as a single term, without a denominator.
pls helpppppp
Simplifying further, you can move any negative exponents to the denominator by changing their sign:
t⁻² u⁻⁶ v⁴ = v⁴ / (t² u⁶)
what is denominator ?
In a fraction, the denominator is the bottom number that represents the total number of equal parts into which the whole is divided. For example, in the fraction 3/4, the denominator is 4, which indicates that the whole has been divided into four equal parts.
In the given question,
When multiplying two terms with the same base, you can add their exponents, so:
t⁻¹ u v⁴* t⁻¹ u⁻⁷ v⁰ = t⁻² u⁻⁶ v⁴
Simplifying further, you can move any negative exponents to the denominator by changing their sign:
t⁻² u⁻⁶ v⁴ = v⁴ / (t² u⁶)
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An employee at a department store is stocking cell phone cases. He has a box of 80 cases. Among the 80 cases, 40 are black, 10 are white, and 30 are pink. If he reaches into the bag randomly and removes one at a time, what is the probability that the first three cases are all pink? Please help I have no clue how to do this at all. I listen in math class but this is hard!
The probability of drawing a pink case on the first draw is 30/80.
What is probability?Probability is a branch of mathematics that deals with the measurement and analysis of random phenomena. It is the study of the likelihood or chance of an event occurring. It involves analyzing and predicting the likelihood of certain outcomes based on the available information and the rules of probability theory. Probability is used extensively in various fields, including statistics, physics, finance, engineering, and computer science, among others.
In the given question,
The probability of drawing a pink case on the first draw is 30/80. Since the cases are not replaced, the probability of drawing a pink case on the second draw is 29/79 (there are now 29 pink cases left out of 79 total cases). Similarly, the probability of drawing a pink case on the third draw is 28/78. Therefore, the probability of drawing three pink cases in a row is:
(30/80) * (29/79) * (28/78) = 0.0412 or about 4.12%.
So, the probability that the first three cases are all pink is 0.0412 or about 4.12%.
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Can anyone please help me answer this question?
Find the limit of p(x)= (x^4 - x^3 - 1)/ x^2 (x + 1) as x approaches -3
Answer:
Step-by-step explanation:To find the limit of p(x) as x approaches -3, we can first simplify the expression by factoring the numerator:
p(x) = (x^4 - x^3 - 1) / x^2(x + 1)
= [(x - 1)(x^3 + x^2 - x - 1)] / [x^2(x + 1)]
Now, when x approaches -3, the denominator of the fraction becomes zero, which means we have an indeterminate form of the type 0/0. To evaluate the limit, we can use L'Hopital's rule, which states that if we have an indeterminate form of the type 0/0 or infinity/infinity, we can take the derivative of the numerator and denominator separately and then evaluate the limit again.
Taking the derivative of the numerator and denominator, we get:
p'(x) = [(3x^2 - 2x - 1)(x^2 + 2x) - 2(x - 1)(2x + 1)] / [x^3(x + 1)^2]
Now, plugging in x = -3 into the derivative, we get:
p'(-3) = [(3(-3)^2 - 2(-3) - 1)((-3)^2 + 2(-3)) - 2((-3) - 1)(2(-3) + 1)] / [(-3)^3((-3) + 1)^2]
= [28 - 44] / [(-3)^3(-2)^2]
= -16 / 108
= -4 / 27
Since the derivative is defined and nonzero at x = -3, we can conclude that the original limit exists and is equal to the limit of the derivative, which is:
lim x->-3 p(x) = lim x->-3 [(x - 1)(x^3 + x^2 - x - 1)] / [x^2(x + 1)]
= p'(-3)
= -4 / 27
Therefore, the limit of p(x) as x approaches -3 is equal to -4/27.
Answer:
[tex]\lim_{x \to -3}p(x) =-\dfrac{107}{18}[/tex]
Step-by-step explanation:
Given the function [tex]p(x)=\dfrac{x^4-x^3-1}{x^2(x+1)}[/tex]
Let's give the expressions in the numerator and denominator their own function names so they are easy to refer to:
n, for numerator: [tex]n(x)=x^4-x^3-1[/tex]
d, for denominator: [tex]d(x)=x^2(x+1)[/tex]
So [tex]p(x)=\dfrac{n(x)}{d(x)}[/tex]
Now, we want the limit of p(x) as x goes to -3.
[tex]\lim_{x \to -3}p(x) =\lim_{x \to -3}\dfrac{n(x)}{d(x)}[/tex]
For limits of quotients, it is important to analyze the numerator and the denominator.
Take a moment to observe that inputting -3 into the denominator is defined and does not equal zero: [tex]d(-3)=(-3)^2((-3)+1)=-18\ne0[/tex]
Also, observe that inputting -3 into the numerator is defined: [tex]n(-3)=(-3)^4-(-3)^3-1=81+27-1=107[/tex]
Importantly, both functions n & d are polynomials, which are functions that are continuous over [tex]\mathbb{R}[/tex].
Since both functions n & d are continuous, both n & d are defined at [tex]x=-3[/tex], and [tex]d(-3)\ne0[/tex], then the limit of the quotient is the quotient of the limits:
[tex]\lim_{x \to -3}\dfrac{n(x)}{d(x)}=\dfrac{ \lim_{x \to -3}n(x)}{ \lim_{x \to -3}d(x)}[/tex]
From here, again, since n & d are continuous over [tex]\mathbb{R}[/tex] and defined at the limit, [tex]\lim_{x \to -3}n(x)}=n(-3)[/tex] and [tex]\lim_{x \to -3}d(x)}=d(-3)[/tex].
Therefore,
[tex]\lim_{x \to -3}p(x) =\lim_{x \to -3}\dfrac{n(x)}{d(x)}=\dfrac{ \lim_{x \to -3}n(x)}{ \lim_{x \to -3}d(x)}=\dfrac{n(-3)}{d(-3)}=\dfrac{107}{-18}=-\dfrac{107}{18}[/tex]