Answer:
Similarly we can convert a limit of the form [infinity]/[infinity] to a limit of the form 0/0 or [infinity]/[infinity] by using the same technique of multiplying numerator and denominator by appropriate factors.
Step-by-step explanation:
To convert a limit of the form 0/[infinity] to a limit of the form 0/0 or [infinity]/[infinity], we can use the following algebraic manipulation:
Multiply the numerator and denominator by the reciprocal of the highest power of the variable in the denominator.
This will usually be the variable that appears in the denominator under a square root, a logarithm, or a trigonometric function.
Simplify the resulting expression by canceling out any common factors.
Evaluate the limit of the simplified expression.
Let's illustrate this with an example:
Example: Find the limit as x approaches infinity of x^2 / (e^x - 1)
Step 1: Multiply numerator and denominator by 1/x^2:
(x^2 / x^2) / [(e^x - 1) / x^2]
Step 2: Simplify the expression by canceling out x^2 in the denominator:
1 / [(e^x - 1) / x^2]
Step 3: Evaluate the limit of the simplified expression. As x approaches infinity, e^x grows faster than x^2, so the denominator goes to infinity and the limit is 0.
Therefore, we have converted the limit of the form 0/[infinity] to a limit of the form 0/0.
Similarly we can convert a limit of the form [infinity]/[infinity] to a limit of the form 0/0 or [infinity]/[infinity] by using the same technique of multiplying numerator and denominator by appropriate factors.
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3. The Lovely Day Fruit company makes two varieties of fruit drink, Fruit Delight and Heavenly Punch. The chart below shows how much juice goes into each type and the profit per can of juice. Each week, the company receives a shipment of 9000 ounces of pineapple juice, 24000 ounces of orange juice, and 14000 ounces of apricot juice. Fill in the question marks appropriately and then use the linear programming process to find how many cans of each should be made each week to maximize profit.
The optimal solution is x = 2000 and y = 4000, meaning that 2000 cans of Fruit Delight and 4000 cans of Heavenly Punch should be made each week to maximize profit.
Fruit Delight:
Pineapple juice: 3 oz
Orange juice: 6 oz
Apricot juice: 2 oz
Profit: $1.50
Heavenly Punch:
Pineapple juice: 6 oz
Orange juice: 2 oz
Apricot juice: 4 oz
Profit: $2.00
Linear Programming Process:
We can use linear programming to solve this problem. We will create two variables, x and y, to represent the number of cans of Fruit Delight and Heavenly Punch respectively. The objective is to maximize profit, which can be expressed mathematically as:
Maximize: 1.50x + 2.00y
subject to:
3x + 6y ≤ 9000
2x + 4y ≤ 14000
6x + 2y ≤ 24000
x ≥ 0
y ≥ 0
Using linear programming software, we can solve this problem and find that the optimal solution is x = 2000 and y = 4000, meaning that 2000 cans of Fruit Delight and 4000 cans of Heavenly Punch should be made each week to maximize profit.
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Determine the type of dilation shown and the scale factor used.
Enlargement with scale factor of 1.5
Enlargement with scale factor of 2.5
Reduction with scale factor of 1.5
Reduction with scale factor of 2.5
The type of dilation and scale factor in the figure shown is
Enlargement with scale factor of 2.5How to find the type of dilationScale factors are used to increase or decrease image. The situation of increment is usually called magnifying or enlargement.
The given problem is enlargement since the preimage is smaller than the image. Also the scale factor will be greater than 1
Taking a point of reference say side 6 in the preimage and side 15 in the image and solve for the factor, assuming the factor is k
6k = 15
k = 15/6
k = 2.5
hence we say that the scale factor is 2.5
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Answer: Enlargement with scale factor of 2.5
Step-by-step explanation: I took the quiz so you don't have too!
Dante deposits $300 to open a savings account. He will earn 2% interest each year on his
money If he makes no deposits or withdrawals, how much money will Dante earn in interest
in 1 year? How much money will be in the account after 4 years?
Therefore , the solution to the given problem of interest comes out to be $324.
What is meant by interest?By dividing the original capital by the cost of borrowing, the time period, and other factors, simple interest is calculated. Simple return is calculated as follows: principal + interest + hours. This method of calculating interest is the simplest. The most typical method is to calculate it as a percent of the principal amount. For instance, if he took $100 from a buddy and promised to repay it with 5% interest, he would only be required to pay his proportion of the interest. $100 (0.05) = $5. When you make loans, you should pay interest, and you also need to when you lend money. Interest is often computed as a percentage of a loan.
Here,
Given: $300 is the principal amount
Time: 4 years,
2% interest rate
To discover the basic interest:
Using this formula
Simple interest equals
=> P*R*T/100,
=> 300*2*4/100
=> 24.
Total after four years equals 300 + 24 = 324
Therefore , the solution to the given problem of simple interest comes out to be $324.
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12. In March, Mrs. Jones's business spent $480 on raw materials. Based on the information in the graph, how much money was spent on business expenses in March? A. $1,600 B. $144 C. $2,100 D. $1,850
By addition, $1,850 is the right response. Mrs. Jones' company spent $1,850 on business expenditures in March, including $480 on raw materials, $1,200 on labour, and $170 on other expenses.
what is addition?The other three fundamental operations are subtraction, multiplication, and division. Addition is one of the four basic operations. The sum or total of these combined values is obtained by adding two integers. The process of merging two or more numbers is known as addition in mathematics. Numbers are added together to form addends, and the outcome of this operation, or the final response, is referred to as the sum. This is one of the crucial mathematical operations we employ on a regular basis. You would add numbers in a variety of circumstances. Combining two or more numbers is the foundation of addition. You can learn the fundamentals of addition if you can count.
Mrs. Jones's business spent $480 on raw materials
Mrs. Jones' company spent $1,850 on business expenditures in March,
$1,200 on labour
$170 on other expenses.
by addition
Total = 1200+480+170 = $1850
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Can someone please solve this math problem? I can't use words, only numbers, infinity symbols, etc... The math problem is in the picture, Thanks!
Answer: 10
Step-by-step explanation:
The table shows how the amount of Ariel’s bank account changed over several weeks. Fill in the missing values.
The missing values from Ariel’s bank account are;
3. -$155.15
4. -$223.75
5. -$1.40
6. -$7.2
What is the change in Ariel’s bank account?Week 1:
Account balance = $360.00
Change from previous week = nil (this is because the previous week account balance is not given)
Week 2:
Account balance = $337.50
Change from previous week = $337.50 - $337.50
= -$22.50
Week 3:
Account balance = $182.35
Change from previous week = $182.35 - $337.50
= -$155.15
Week 4:
Account balance = -$41.40
Change from previous week = -$41.40 - $182.35
= -$223.75
Week 5:
Account balance = -$1.40
Change from previous week = -$1.40 - (-$41.40)
= $40.00
Week 6:
Account balance = -$5.80
Change from previous week = -$5.80 - (-$1.40)
= -$7.2
Therefore, the change in Ariel's bank account from previous week can be calculated by finding the difference between the present and previous week account balance.
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) Select the inequality that the number line represents. Use x as the variable.
The number line represents an inequality. The inequality is x ≥ -3.
What is a number line?
A number line is a picture of a graduated straight line that serves as a visual representation of real numbers in primary mathematics. Every number line point is considered to correspond to a real number and every real number to a number line point.
If a number line contains a close circle, then the number includes in the inequality.
If a number line contains an open circle, then the number does not include in the inequality.
In the graph, there is a close circle on -3.
The line represents all real numbers greater than -3.
Since there is a close circle on -3, thus -3 include in the inequality.
The given variable is x.
The graph represents all real numbers greater than or equal to -3.
The inequality is x ≥ -3.
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Which point is the best approximation of the relative maximum of the polynomial function graphed below? A. -2.2,15 B. -3.6,17 C. -2.7,16 D. -1,8,15.
Option B is the correct answer. The best approximation of the relative maximum of the polynomial function graphed below is (-3.6,17)
The relative maximum is in the peak on the left side.
A function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, or cubic equation.
The general Polynomial Formula is
[tex]P(x)=a_{n} x^n+a_{n-1} x^n^-1+..........+a_{2} x^2+a_{1} x^1+a_{0}[/tex]
where all the powers are non-negative integers.
And, a0,a1,………,an ∈ R
The graph of function depends upon its degree, and having one variable which has the largest exponent is called a degree of the polynomial.
There are some Graphs of Higher Degree Polynomial Functions:
The Standard form is:
P(x) = an xn + an-1 xn-1+.……….…+ a0
an, an-1, … a0 are real number constantsan n can’t be equal to zero and is called the leading coefficientn is a non-negative integerEach exponent of a variable in the polynomial function should be a whole numberTo know more about the Polynomial function:
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What is the solution to the system of equations y equals one over 3x 10?
The solution to the given system of equations y = 1/3x - 10 and 2x + y = 4 is (6, -8).
y = 1/3x - 10 … (1)
2x + y = 4 … (2)
We can use the substitution method to determine the solution of the given system of linear equations.
Substituting (1) in (2), we get
⇒2x + (1/3 x - 10) = 4
⇒2x + 1/3 x - 10 = 4
⇒(6 + 1)/3 x = 14
⇒7/3 x = 14
⇒7x = 14 × 3
⇒7x = 42
Dividing both sides by 7
x = 6
Place the value of x in equation (1)
⇒y = 1/3 (6) - 10
⇒y = 2 - 10
⇒y = -8
--The given question is incomplete, the complete question is
"What is the solution to the system of equations y = 1/3x - 10 and 2x + y = 4?"--
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From midnight until 8:00 AM, the temperature tripled and then rose 5 degrees. The temperature at 8:00 AM was –7 degrees Celsius. Write and solve an equation to find the temperature at midnight.
The equation of the temperature at midnight is written as 3y + 5 = -7
The temperature at the midnight is calculated to be -4 degrees
How to write the equation of the temperature at midnightThe equation for the temperature at midnight is following the information given in the problem tripled and increased by 5
Assuming the temperature at midnight is y. For the temperature to triple we have
3y
For the temperature to rise 5 degrees will result to
3y + 5 = -7
solving for y
3y + 5 = -7
3y = -7 - 5
3y = -12
y = -4
At midnight the temperature was -4 degrees
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i need a solution and where its graphed please help
Case 1: The linear equations has infinite solutions.
Case 2: The linear equations has no solution.
How to find the solution of a system of linear equations graphically
In this problem we find two cases of systems of two linear equations with two variables, which are described by a system of the form:
a · x + b · y = c
d · x + e · y = f
Where:
a, b, d, e - Dependent variable.c, f - Independent variable.The solution of this kind of systems can be found graphically on Cartesian plane, whose procedure is described briefly:
Determine x-intercept on each line.Determine y-intercept on each line.Mark the intercepts on Cartesian plane.Construct the resulting lines. Mark the point of intersection of the two lines.Now we proceed to find the solution for each case:
Case 1
Line 1: y = 0.5 · x + 1
y-Intercept
y = 0.5 · 0 + 1
y = 1
(x, y) = (0, 1)
x-Intercept
0 = 0.5 · x + 1
0.5 · x = - 1
x = - 2
(x, y) = (- 2, 0)
Line 2: - 2 · x + 4 · y = 4
y-Intercept
- 2 · 0 + 4 · y = 4
4 · y = 4
y = 1
(x, y) = (0, 1)
x-Intercept
- 2 · x + 4 · 0 = 4
- 2 · x = 4
x = - 2
(x, y) = (- 2, 0)
Since the two lines have the same intercepts, then there are infinite solutions.
Case 2
Line 1: y = - x - 3
y-Intercept
y = - 0 - 3
y = - 3
(x, y) = (0, - 3)
x-Intercept
0 = - x - 3
x = - 3
(x, y) = (- 3, 0)
Line 2: y + x = 2
y-Intercept
y + 0 = 2
y = 2
(x, y) = (0, 2)
x-Intercept
0 + x = 2
x = 2
(x, y) = (2, 0)
The system of linear equations has no solution as both lines are parallel.
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How to prove 3 points are collinear in 3d using section formula?
The section formula provides a way to check if three points in 3D space are collinear by verifying that the equation (x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1) = 0 is satisfied when their coordinates are substituted.
To prove three points are collinear in 3D using the section formula, first determine the coordinates of the three points. Then, substitute the coordinates into the section formula equation, (x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1) = 0. Finally, calculate the equation and if it is equal to 0, the three points are collinear.
For example, if the coordinates of three points are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3), then the three points are collinear if the following equation is satisfied:
(x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1) = 0
If the equation is not equal to 0, the points are not collinear. To illustrate, if the coordinates of three points are
(x1, y1, z1), (x2, y2, z2) and (x3, y3, z3),
then the three points are collinear if the following equation is satisfied: (x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1) = 0.
If this equation is not equal to 0, then the three points are not collinear. To check if the points are collinear, one must calculate the equation and determine if the equation is equal to 0. If so, the three points are collinear.
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If the quadrilateral ABCD is reflected across the x-axis, what will be the coordinates of B’?
The coordinate of the point reflection across the x-axis will be (3, -6). Then the correct option is A.
What is a transformation of a point?A spatial transformation is each mapping of feature space to itself and it maintains some spatial correlation between figures.
The reflection does not change the shape and size of the geometry. But flipped the image.
The quadrilateral ABCD is shown. The coordinate of point B is (3, 6).
If the quadrilateral ABCD is reflected across the x-axis. Then the coordinates of B' will be given as,
B' = (3, -6)
The coordinate of the point reflection across the x-axis will be (3, -6). Then the correct option is A.
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Enter "100 divided by the sum of 22 and 3" as an expression.
Answer:
this question has 3 answers, according to improper fraction your answer can be 50 over 11(50/11).
According to decimal point your answer can be 4.55.
Step-by-step explanation:
According to mixed numbers your answer can be 4 as a whole number 6as numerator and 11 as denominator
Answer:
100÷(22+3)
Step-by-Step:
you cant leave it as 100÷22+3 because pemdas states that division would be first but according to the problem it needs to be the last part. parenthesis comes before any other operation so the only way to do addition first is to put it in parenthesis.
1. Two polygons are similar. If one side of the first polygon is twice the length of the corresponding side of. the second.what is the ratio of their perimeter?
Answer:I don’t know I just need points
Step-by-step explanation:
4^(3x)= 16^(x-2)
Solve for x.
Answer:
x = -4
Step-by-step explanation:
4^(3x) = 16^(x-2)
First, we create equivalent expressions in the equation that all have equal bases.
4^3x = 4^2(x-2)
Since the bases are the same, two expressions are only equal if the exponents are also equal.
3x = 2(x − 2)
3x = 2x -4
x = -4
So, the answer is x = -4
What is the formula to classify triangles?
The law of cosines serves as a generalization of the Pythagorean formula, which is applicable to all triangles.
How do you determine if a triangle is right acute or obtuse?Compare the total of the squares on the two smaller sides to the square on the largest side to determine if the triangle is acute, right, or obtuse. Because of the larger sum, the triangle is sharp.
The Pythagorean formula applies to all triangles and is generalised by the law of cosines. The square of one side of the triangle, c², is said to be equal to a² + b², the sum of the squares of the other two sides, minus 2ab cos C, which is the product of the two squares times the cosine of the opposite angle.
Three closed sides make up a triangle. The Pythagoras theorem and the Heron's formula are two significant formulas that pertain to triangles.
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Which correlation coefficient indicates a strong positive correlation?
A. r = +0.5
B. r= -0.1
C. r= -0.09
D. r= +0.9
The correlation coefficient r = +0.9 indicates a strong positive correlation as a strong positive (negative) linear relationship is shown by values between 0.7 and 1.0 (or -0.7 and -1.0).
What is corelation coefficient?
A statistical concept known as the correlation coefficient aids in establishing a relationship between expected and actual values gained through statistical experimentation. The estimated correlation coefficient's value explains how well the expected and actual values match.
The following points are the accepted guidelines for interpreting the correlation coefficient -
1. Zero means there is no linear relationship.
2. A perfect positive linear relationship, or one with a value of 1, means that as one variable rises in value, the other variable rises in value according to an exact linear law.
3. A perfect negative linear relationship is represented by a value of 1; as one variable's values rise, the other variable's values fall according to an exact linear law.
4. Values between 0 and 0.3 (0 and -0.3) denote a wobbly linear rule with a weak positive (negative) linear relationship.
5. A fuzzy-firm linear rule indicates a somewhat positive (negative) linear relationship for values between 0.3 and 0.7 (-0.3 and -0.7).
6. A strong positive (negative) linear relationship is shown by values between 0.7 and 1.0 (or 0.7 and 1.0) by a strict linear rule.
Therefore, r = +0.9 indicates a strong positive correlation.
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Chritine' Deli i making hero andwiche for the upcoming football party. Chritine ha 262 piece of bread, 336 lice of patrami, and 302 lice of cheee. If he need to make hero each coniting of two piece of bread, two lice of patrami and two lice of cheee, what i the greatet number of hero he can make?
Christine has 262 pieces of bread, 336 lice of patrami, and 302 lice of cheese, so she can make a maximum of 131 hero sandwiches if each sandwich consist of two pieces of bread, two slices of patrami, and two slices of cheese.
Therefore the answer is 131.
Christine is making hero sandwiches, each of which consist of two pieces of bread, two slices of patrami, and two slices of cheese. To find the greatest number of heroes he can make, we need to divide the total number of each ingredient by the number of that ingredient used in each sandwich.
The number of heroes Christine can make with the bread is:
262 pieces of bread / 2 pieces of bread per hero = 131 heroes
The number of heroes Christine can make with the patrami is:
336 slices of patrami / 2 slices of patrami per hero = 168 heroes
The number of heroes Christine can make with the cheese is:
302 slices of cheese / 2 slices of cheese per hero = 151 heroes
To find the greatest number of heroes Christine can make, we need to take the smallest of these three values, which is 131 heroes. This means that Christine can make a maximum of 131 heroes with the ingredients he has on hand.
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Using the numbers 1-9, at most one time each, fill the blanks to make this equation true. Find one solution for this equation.
Answer:
Step-by-step explanation:
(3^1)^4
By using the numbers 3, 1, and 4, you are able to get the answer 81
You are measuring the height of a spruce tree. You stand 45 feet from the base of the
tree. You measure the angle of elevation from the ground to the top of the tree to be
59. Find the height h of the tree to the nearest foot.
Provided the given distance and the angle of elevation, Using the trigonometry in the right angled triangle formed, The answer is 74.89 feet.
What is trigonometry?Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. Trigonometry is used to study properties such as lengths, angles, and areas of triangles and the relationships between them. It is also used in many fields, including physics, engineering, and navigation. Trigonometry is based on the study of the ratios of the sides of a right triangle. The three main ratios are sine, cosine, and tangent. These ratios are defined using the right triangle's angles and are often abbreviated as sin, cos, and tan. In addition to these three main ratios, there are also several other trigonometric functions, including cosecant, secant, and cotangent. These functions are the reciprocals of the sine, cosine, and tangent functions, respectively.
What is a right triangle?A right triangle is a type of triangle that has one angle that measures exactly 90 degrees. This angle is called the right angle and the two sides that form the angle are called the legs of the triangle. The side opposite the right angle is called the hypotenuse. A right triangle can also be defined as a triangle that has a 90-degree angle and two acute angles. The legs of the triangle are the sides that are opposite the acute angles, and the hypotenuse is the side that is opposite the right angle.
tan(A) = rise/run
tan(59) = height/45
height = tan(59)*45
=74.89 feet
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John i 7 year younger than jane. The um of their age i 10 year ago wa 55 year. How old are they now
John 38 years old and Jane 45 years old 10 years ago from the present age .
John: 48 years old
Jane: 55 years old
Jane is 55 years old and John is 7 years younger, so John is 55 - 7 = 48 years old.
John is 7 years younger than Jane, so if we subtract 7 years from Jane's age, we can find John's age. Jane is 55 years old, so if we subtract 7 years, we get 55 - 7 = 48. Therefore, John is 48 years old, and Jane is 55 years old. To find their age 10 years ago, we can subtract 10 years from each of their current ages. This would make John 38 years old and Jane 45 years old 10 years ago from the present age
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What is y in this equation?
Answer:
y = 15.4
Step-by-step explanation:
the outer and inner triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{y}{11}[/tex] = [tex]\frac{8+3.2}{8}[/tex] = [tex]\frac{11.2}{8}[/tex] ( cross- multiply )
8y = 11 × 11.2 = 123.2 ( divide both sides by 8 )
y = 15.4
The volume of each cube is 1/64ins^3 Multiply the number of cubes needed to fill the prism by the volume of each cube to find the volume of the prism. Include units.
4. Instead of using cubes, Kaia uses the formula V = 1 x w × h. Fill in the spaces bel to complete her work. Include units.
V=lxwxh
V=
in3
Volume of a cube = a3
here a = 4
then = a3 = 4^3 =64
The rectangular prism shows 12 cubes: 3 rows and 4 columns.
The volume of a prism is the area of the base times the height,
The volume of a rectangular prism is the length times width times height,
The volume of a cube is the length of one side cubed,
Volume of a rectangular prism = (length x width x height) cubic units.
V = (l x w x h) cubic units
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How many longitudes are there class 9?
There are 180 degrees of longitude, which is divided into 360 lines of longitude (meridians).
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is measured in degrees, with a full circle consisting of 360 degrees. The Prime Meridian is located at 0 degrees longitude, and each degree is further divided into 60 minutes and each minute is divided into 60 seconds. Therefore, there are 180 degrees of longitude east of the Prime Meridian and 180 degrees of longitude west of the Prime Meridian.
There are 180 degrees of longitude, which is divided into 360 lines of longitude (meridians).
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Question 10 of 22
What is the solution to this equation?
x-14=-9
A. x = 23
B. x = 5
C. x = -5
OD. x = -23
SUBINT
Triangles S, M, and L are scaled copies of one another. Determine the area of each triangle.
The area of each triangle is given as follows:
Triangle S: 4 units squared.Triangle M: 16 units squared.Triangle L: 36 units squared.How to obtain the area of a triangle?The area of a triangle is given by half the multiplication of the base and the height, as follows:
Area = 0.5 x base x height.
The dimensions for each triangle are given as follows:
Triangle S: base 2, height 4.Triangle M: base 4, height 8.Triangle L: base 6, height 12.Hence the areas are:
Triangle S: 0.5 x 2 x 4 = 4.Triangle M: 0.5 x 4 x 8 = 16.Triangle L: 0.5 x 6 x 12 = 36.Missing InformationThe triangles are given by the image presented at the end of the answer.
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Which is the discriminant of the equation x2 =-- 4x 4?
Answer: The discriminant of a quadratic equation ax² + bx + c = 0 is given by the formula:
b² - 4ac
Given the equation x^2 - 4x - 4 = 0, we can find the discriminant by substituting the values of a, b, and c into the formula:
b² - 4ac = (-4)² - 4(1)(-4) = 16 + 16 = 32
So, the discriminant of the equation x^2 - 4x - 4 = 0 is 32.
Step-by-step explanation:
Alex made a table to record how many trees were in his yard for each day that the landscapers were working. Which equation represents the data?
A tent is shown in the figure. If the amount of air only in cylindrical part is 300m^3, calculate the volume of the air in the tent.
Answer:
400 m³
Step-by-step explanation:
The tent consists of a cylinder at the bottom and the roof of the tent is a cone
The height of the cone = 2h - h = h which is the same as the height of the cylindrical part
Volume of a cylinder = πr²h where r = radius, h- height
We are given this volume = 300 m³
The volume of a cone is (1/3)πr²h
If the heights and radius are the same, the volume of the cone is 1/3 the volume of the cylinder
Volume of conical part of tent = 1/3 x volume of cylindrical part
= 1/3 x 300 = 100 m³
Total volume of the tent and therefore the volume of air
= 300 + 100 = 400 m³