Answers
Related Questions
A box exerts a force of 10,000 N over an area of 1 m2. What pressure is the box exerting on the floor?
Answers
Answer:
The area is in contact with the ground if A box exerts 10,000 Pa of pressure on the ground. If the box weighs 1000 N is 10 m².
( which is D/10 m²).
Step-by-step explanation:
so what is pressure?
In the physical sciences, pressure is defined as the stress at a point within a confined fluid or the perpendicular force per unit area. A 42-pound box with a bottom area of 84 square inches will impose pressure on a surface equal to the force divided by the area it is applied to, or half a pound per square inch.
Answer: 10,000 pa
Step-by-step explanation:
Pressure = Force / Area
Force = 10,000
Area = 1 m^2
Pressure = 10,000 / 1
Pressure = 10,000 pa.
Nathan and some friends are going to the movies. At the theater, they sell a bag of popcorn for $4.50 and a drink for $4.25. How much would it cost if they bought p bags of popcorn and d drinks?
Answers
If they purchased 7 bags of popcorn and 3 drinks, the price would be $44.25; p bags of popcorn and d beverages would cost 4.50p+4.25d.
What is a formula?
Equation is the name given to two or more expressions with the equal sign.
Due to that
$4.50 worth of popcorn
beverage for $4.25.
There are 3 drinks and 7 bags of popcorn.
So let's calculate the overall cost.
7(4.50)+3(4.25)
31.5+12.75
44.25
So if they purchased 7 bags of popcorn and 3 beverages, the price would be $44.25
The price for p bags of popcorn and d drinks must now be determined.
4.50p+4.25d
Therefore, the price would be $44.25 if they purchased 7 bags of popcorn and 3 drinks, and the cost would be 4.50p+4.25d for p bags of popcorn and d beverages.
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Which example illustrates the associative property for addition?
(3 x 7) + 8 = 3+ (7 x 8) = (3x 8) + 7
(3+7) x 8 = 3 x (7 + 8) = (3 + 8) x 7
(3+ 7) + 8 = 3x (7 x 8) = (3 + 8) + 7
(3 + 7) + 8 = 3 + (7 + 8) = (3 + 8) + 7
Answers
Answer:
The answer is that the third example illustrates the associative property for addition. The associative property states that the order in which numbers are added does not affect the result. In other words, (a + b) + c = a + (b + c) for all numbers a, b, and c.
The third example follows this pattern:
(3 + 7) + 8 = 3 + (7 + 8) = (3 + 8) + 7
The other examples do not follow this pattern, so they do not illustrate the associative property for addition.
Step-by-step explanation:
The third example illustrates the associative property for addition. The associative property states that the order in which numbers are added does not affect the result. In other words, (a + b) + c = a + (b + c) for all numbers a, b, and c.
The third example follows this pattern:
(3 + 7) + 8 = 3 + (7 + 8) = (3 + 8) + 7
The other examples do not follow this pattern, so they do not illustrate the associative property for addition.
The first example illustrates the associative property for multiplication, which states that the order in which numbers are multiplied does not affect the result. In other words, (a x b) x c = a x (b x c) for all numbers a, b, and c.
The second example is not a valid mathematical expression, as it attempts to add a number and a product.
Answer:
(3 + 7) + 8 = 3 + (7 + 8) = (3 + 8) + 7
Step-by-step explanation:
The equations for associative property for addition are:
(a + b) + c = a + (b + c)
So, (3 + 7) + 8 = 3 + (7 + 8) = (3 + 8) + 7 is the answer!
in a survey, 32 people were asked how much they spent on their child's last birthday gift. the results were roughly bell-shaped with a mean of $48 and standard deviation of $7. construct a confidence interval at a 80% confidence level. give your answers to one decimal place.
Answers
The required 80% confidence interval is (46.15, 49.85).
What is a confidence interval?In frequentist statistics, a confidence interval is a range of estimates for an unknown quantity.
The most common confidence level is 95%, but when calculating confidence intervals, other levels, such as 90% or 99%, are also occasionally employed.
So, we know that:
n = 32
χ' = 48
s = 8
Create an 80% confidence interval using the equation below:
[tex]\left(\bar{X} \pm t_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}}\right)[/tex]
To determine the critical value t for n-1 degrees of freedom and 1 -cl level of significance, use the formula below:
[tex]t_{0.2, d f=31}=\pm 1.309[/tex]
Create the confidence interval now:
[tex]\begin{aligned}& \left(48 \pm 1.309 \times \frac{8}{\sqrt{32}}\right) \\& (48 \pm 1.85) \\& (46.15,49.85)\end{aligned}[/tex]
Therefore, the required 80% confidence interval is (46.15, 49.85).
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(Multiplying Linear Expressions MC)
Simplify −5g(3g + 4).
A: −15g + 4
B: −15g − 20g
C: −15g2 + 4
D: −15g2 − 20g
Answers
The multiplication of the linear expression - 5g(3g + 4) is given by -15g² - 20g.
The correct answer option is option D
How to multiply linear expression?Multiplication: This is the process of computing the sum of a number with itself a specified number of times, or any other analogous binary operation that combines other mathematical objects.
- 5g(3g + 4)
open parenthesis
= -5g × 3g - 5g × 4
= -15g² - 20g
In conclusion, the linear expression - 5g(3g + 4) when simplified equals -15g² - 20g
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what is the measure of X
Answers
The measure of an angle X is 11° and 3X+2°=35° , 4X-9° =35° .
What is meant by an angle?In Euclidean geometry, an angle is a figure made up of two rays that have a shared vertex, also known as a common terminus, and are referred to as the angle's sides. The angles of two rays are situated in the plane containing the rays. An angle is also created when two planes intersect at an angle. These are referred to as dihedral angles. Another technique to define two curves is to specify the angle generated by rays that are tangent to the intersection of two curves. Angle is another word for how long a rotation or angle is.
∠QPA = ∠UPA
3X+2 =4X- 9
4X-3X= 9 + 2
X= 11
The measure of X is 11
3X+2°=3(11)+2°
=35°
(4X-9)°=(4(11)-9)°
=35°
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3(x-7)+12=1/4(12x-8)-7
Answers
Answer:
What's the question? Ill try to answer in comments
Step-by-step explanation:
7. When the height of the sand in a particular rectangular sandbox is
leveled out then the height of the sand, in inches (in.), is proportional to
the volume of sand, in cubic inches (in.3), in the sandbox. When the
height of the sand is 1.25 in. the volume of the sand is 280 in.3. A
playground has 3 of these sandboxes.
What is the total volume of the sand, in cubic inches (in.3), that is
needed for the playground when the height of the sand in each sandbox
is 4.5 in.?
Show your work in the provided space.
Answers
The Total volume of sand in cubic inches needed for the playground when the height of the sand in each sandbox is 4.5 in is; 2700 in³
How to solve mathematical proportions?
We are told that the height of the sand, in inches (in.), is proportional to
the volume of sand.
Let the volume be denoted as V
Let the height be denoted as h
Thus;
V ∝ h
Then to equate this, we will have a constant of proportionality;
V = kh
where k is constant of proportionality
When the height of the sand is 1.25 in., the volume of the sand is 280 in³ and this gives;
280 = 1.25k
k = 250/1.25
k = 200
Now, ware told that the height of sand in a box is now 4.5 inches and so;
V = 200 * 4.5
V = 900 in³
Since the playground has 3 of such boxes, then we can say that;
Total volume of sand on the playground = 900 * 3 = 2700 in³
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what is the translation rule
Answers
Answer:
A translation is a type of transformation that moves each point in a figure the same distance in the same direction
Step-by-step explanation:
Translations are often referred to as slides. You can describe a translation using words like "moved up 3 and over 5 to the left" or with notation.
The sum of three numbers is 24. The third is 11 less than 3 times the second. 8 times the first is 4 less than 10 times the second. Find the numbers
Answers
Answer:
F + S + T = 24 F= 1st number, S = 2nd, T = 3rd
Step-by-step explanation:
T = 2S - 11
9F = 7 + 2S
divide by 9
F + (7+2S)/9
substitute for F and T
(7+2S)/9 + S + 2S -11 = 24
7+2S + 9S + 18S - 99 = 216
29S = 216 + 99 - 7 = 308
S = 308/29 = the 2nd number
T = 2S -11 = 616/29 -11 = 616/29 - 319/29 = 297/29 = 3rd number
F = (7+2S)/9 = (7 +616/29)/9 = 819/261 = 1st number
the 3 numbers are 819/261, 308/29 and 297/29
they sum to 819/261 + 308/29 + 297/29 = about 3.14 + 10.62 +10.24 = 24
F =( 7+2(10.62))/9 = (7+ 21.24)/9 = 28.24/9 = 314
S = 308/29 = 10.62
T = 2S-11 = 2(10.62) - 11 = 21.24-11 = 10.24
although odds are the problem may have been mis-copied or has a slight typo, with more integer type solutions in the corrected version. If the problem had an 8 instead of 9, (8F = 7+ 2S) then F=3.5, S=10.5, T = 10 as exact answers, not rounded off. 3.5+10.5+ 10 = 24
First number: [tex]\frac{167}{21}[/tex];
Second number: [tex]\frac{142}{21}[/tex];
Third number: [tex]\frac{130}{14}[/tex].
Step-by-step explanation:1. Assign variables to each number.First number: "x";
Second number: "y";
Third number: "z".
2. Form equations based on the problem's statement.Equation 1. "The sum of three numbers is 24", hence:
[tex](1)x+y+z=24[/tex]
Equation 2. "The third is 11 less than 3 times the second.", hence:
[tex](2)z=3y-11\\ \\(2)-3y+z=-11[/tex]
Equation 3. "8 times the first is 4 less than 10 times the second.", hence:
[tex](3)8x=10y-4\\\\(3)8x-10y=-4[/tex]
3. Group the 3 equations.[tex](1)x+y+z=24\\\\(2)-3y+z=-11\\\\(3)8x-10y=-4[/tex]
4. Expand the equations.As you may see, the 3 variables don't always appear on all 3 equations. Therefore, we'll have to introduce them even though they don't appear. For this, we write the variable with a coefficient of 0 next to it
[tex](1)x+y+z=24\\\\(2)0x-3y+z=-11\\\\(3)8x-10y+0z=-4[/tex]
5. Rewrite the equations as a 3x4 matrix.Using the coefficient of each variable in each equation, rewrite the system of equation into a matrix like this:
[tex]\left[\begin{array}{cccc}1&1&1&24\\0&-3&1&-11\\8&-10&0&-4\end{array}\right][/tex]
6. Convert this matrix into the row-echelon form.Check the attached image to see the steps to making this convertion.
a) Getting the 1 on column 1.
Since there's already a 1 there, we skip this step.
b) Getting the 0 on column 1.
Since there's already a 0 there, we skip this step.
c) Getting the 0 on column 1.
Multiply row 1 values by "-8" and add them to row 3. The result of these operations should be:
[tex]\left[\begin{array}{cccc}1&1&1&24\\0&-3&1&-11\\0&-18&-8&-196\end{array}\right][/tex]
d) Getting the 1 on column 2.
Since we are working with column 2 and row 2, you may use the pivot value "-3" (the one that corresponds to the intersection of column 2 and row 2) and divide row 2 by itself to obtain a 1. Result is the following:
[tex]\left[\begin{array}{cccc}1&1&1&24\\0&1&-\frac{1}{3} &\frac{-11}{-3} \\0&-18&-8&-196\end{array}\right][/tex]
e) Getting the 0 on column 2.
Multiply row 2 by "18" and add it to row 3. Resulting table is:
[tex]\left[\begin{array}{cccc}1&1&1&24\\0&1&-\frac{1}{3} &\frac{-11}{-3} \\0&0&-14&-130\end{array}\right][/tex]
f) Getting the 1 on column 3.
We are working with column 3 and row 3, the intersection value is "-14" and we may use it as a pivot value. Hence, we may divide row 3 by "-14" in order to obtain the number 1 in column 3. Final table is:
[tex]\left[\begin{array}{cccc}1&1&1&24\\0&1&-\frac{1}{3} &\frac{-11}{-3} \\0&0&1&\frac{-130}{-14}\end{array}\right][/tex]
7. Calculate the values.Starting from the bottom up, take the expression of the resulting matrix and calculate the values of each variable.
a) From row 3 we have:
[tex]0x+0y+z=\frac{-130}{-14} \\ \\z=\frac{130}{14}\\ \\[/tex]
b) From row 2 we have:
[tex]0x+y-\frac{1}{3} z=\frac{-11}{-3} \\ \\y-\frac{1}{3} z=\frac{11}{3}\\ \\y-\frac{1}{3} (\frac{130}{14} )=\frac{11}{3}\\ \\y-\frac{130}{42} =\frac{11}{3} \\ \\y=\frac{11}{3}+\frac{130}{42}\\\\y=\frac{11*14}{3*14}+\frac{130}{42}\\ \\y=\frac{154}{42}+\frac{130}{42}\\ \\y=\frac{284}{42} \\\\y=\frac{142}{21}[/tex]
c) From row 1 we have:
[tex]x+y+z=24\\ \\x+\frac{142}{21}+\frac{130}{14} =24\\ \\x=24-\frac{142}{21}-\frac{130}{14}\\ \\x=\frac{1008}{42} -\frac{284}{42} -\frac{390}{42} \\ \\x=\frac{167}{21}[/tex]
8. Verify that the numbers work correctly in each of the equations.Equation 1: [tex]\frac{167}{21} +\frac{142}{21} +\frac{130}{14} =24[/tex] Correct.
Equation 2: [tex]-3(\frac{142}{21} )+\frac{130}{14} =-11[/tex] Correct.
Equation 3: [tex]8(\frac{167}{21} )-10(\frac{142}{21} )=-4[/tex] Correct.
9. Conclude.First number: [tex]\frac{167}{21}[/tex];
Second number: [tex]\frac{142}{21}[/tex];
Third number: [tex]\frac{130}{14}[/tex].
Important: Check the attached Excel sheet to see the changes made in the matrix.
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3 miles is the same as how many kilometers?
Hint: 1 mi≈ 1.6 km
Round your answer to the nearest tenth.
Answers
Answer: 4.8
Step-by-step explanation:
Fill in the blanks 30 = 27 __ ___x 9 = 54 81 / ___ = 9 2 = ___ / 5 8 x ___ = 32 72 / ___ = 8 7* __ = 49
Answers
Step-by-step explanation:
this is a college level question, and you need help with that ? elementary school students have to be able to answer this.
30 = 27 + 3
6 × 9 = 54
81 / 9 = 9
2 = 10 / 5
8 × 4 = 32
72 / 9 = 8
7 × 7 = 49
which of the following is false about probability distributions? O point the probabilities must total O. each probability should be greater than or equal to O. each probability should be positive, less than or equal to O. the outcomes listed must be independent.
Answers
The outcomes listed must be independent is false about probability distributions.
In probability theory, statistics, and the theory of stochastic processes, independence is a key concept. Informally speaking, two occurrences are independent, statistically independent, or stochastically independent if their occurrence has no bearing on either their chances or probability of occurring. Similarly to this, two random variables are independent if the probability distribution of either one is unaffected by the realization of the other.
It is important to distinguish between two definitions of independence when working with collections of more than two occurrences. Informally, mutual independence of events refers to each event being independent of any combination of other events in the collection, whereas pairwise independence of events refers to any two events in the collection being independent of each other.
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The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity ten times that of the other, are 7 m apart. On the line between the two light sources, how far from the stronger light is the total illumination least?
Answers
Answer:
The intensity of the light is inversely proportional to the square of the distance from the source of light. Therefore, in this situation, where one light source has an intensity ten times that of the other, the distance from the stronger light source would be 0.7m before the total illumination is the least.
Step-by-step explanation:
Please help please and thank you.
Answers
Answer: 23 / 40 or 23:40
Step-by-step explanation:
Width = 69
Length = 120
so ratio of width is to length = 69 : 120
simplest form is 23 : 40 or 23 / 40 (fraction)
work out and simplify where possible 9/10 - 7/10
Answers
Answer:
1/5
Step-by-step explanation:
(9-7)/10
2/10
(2:2) / (10:2)
1/5
13. a) (6 pts total) The same series of books you've been wanting cost $44 at Barnes and Noble and are on sale for $35
at Target. If you have a 25% off coupon for Barnes and Noble and a 5% RedCard discount at Target, find what the
price before tax would be at each of the two retailers.
Answers
Answer:
The price before tax at Barnes and Noble is $33, and the price before tax at Target is $33.25.
Step-by-step explanation:
A bag contains 3 gold marbles, 7 silver marbles, and
28 black marbles. Someone offers to play this game:
You randomly select one marble from the bag. If it
is gold, you win $3. If it is silver, you win $2. If it is
black, you lose $1.
What is your expected value if you play this game?
Answers
To calculate the expected value of playing this game, we need to multiply the value of each possible outcome by its probability of occurring and then add these values together. In this case, there are 3 gold marbles, 7 silver marbles, and 28 black marbles in the bag, for a total of 38 marbles. So, the probability of selecting a gold marble is 3/38, the probability of selecting a silver marble is 7/38, and the probability of selecting a black marble is 28/38. The value of winning $3 if a gold marble is selected is $3 * (3/38) = $0.21. The value of winning $2 if a silver marble is selected is $2 * (7/38) = $0.37. And the value of losing $1 if a black marble is selected is -$1 * (28/38) = -$0.74. Adding these values together, we get $0.21 + $0.37 - $0.74 = -$0.16. Therefore, the expected value of playing this game is approximately -$0.16.
What is the value of p?
p=?°
Answers
Answer:
p = 82°
Step-by-step explanation:
Since Triangle HIJ is an isosceles triangle, the base angles of the triangle are equal.
Angle IHJ = 180 - 131
= 49° (sum of angles on a straight line)
Angle IJH = Angle IHJ = 49°
p = 180 - 49 - 49
= 82° (sum of angles in a triangle)
Let the set be defined as follows. A= 5, 8, 34, 59, 73, 79,89. Find the total number of proper subsets of A. Find the total number of subsets of A.
Answers
Answer:
Step-by-step explanation: As we know the number of elements in set A are 7 and the formula to calculate the number of proper subsets is 2^n – 1
Therefore substituting the values in formula . n is number of elements of set A. Here n=7
so 2^7-1= 128-1=127
Ans:127
Two elephants are being delivered to the zoo. One elephant weighs 23,453 pounds, and the other elephant weighs 19,916 pounds. How many pounds do the elephants weigh together?
Answers
The required weight of the two elephants together is 43369 pounds.
How to add two similar quantity?Addition is an approach to consolidating things and considering them all together.Addition in math is a course of consolidating at least two numbers. Addends are the numbers added, and the outcome or the last response we get after the interaction is known as the total.
According to question:We have,
One elephant weighs 23,453 pounds, and the other elephant weighs 19,916 pounds.
To find total weight of two elephant together,we have to add them.
23,453 pounds + 19,916 pounds
43369 pounds
Thus, required weight of two elephant is 43369 pounds.
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Given an ABCD inscribed quadrilateral, where the AB side is the diagonal of the circum- circle of the quadrilateral. BC = 3cm, CD = 5 cm and BCDZ = 120°. Give the length of the BD diagonal, AB and AD sides and the other angles.
Answers
Answer:
Step-by-step explanation:
Given an ABCD inscribed quadrilateral, where the AB side is the diagonal of the circum-circle of the quadrilateral, we can calculate various properties by using basic trigonometry. Firstly, let us determine the length of BD. Since BCDZ = 120° and BC = 3cm ,we can use Sine rule to find out BD which will be equal to 4 cm. Secondly, we need to find AB and AD sides lengths as well as other angles in order for our calculations to be complete. To do this we will use Cosine rule since all three sides are known: BC=3cm; CD=5cm;BD=4 cm . This gives us a value for angle CBD which is approximately 39° and consequently angle BAD is also 39° since they add up together (BAD+CBD)to 180 degrees due their being opposite each other on a straight line..Finally ,using cosine again with these new values gives us both AB(6)and AD(2).
To summarise : Lengths -AB: 6 cm ; BD : 4 cm ;AD 2CM Angles - BCDZ :120 ° ; CBD & BAD :39 °
In conclusion , given an ABCD inscribed quadrilateral whose one side was already identified as its circumference diameter it was possible through simple trigonometric equations such s Sines Rule or Cosines Rule determine its remaining lengths ans angles accurately .
5 triangles are shown. Triangles 1 and 4 are identical. Triangle 2 has identical side lengths and angle measures but is rotated. Triangle 3 has a smaller base than triangles 1, 2, and 4. Triangle 5 is double the size of triangles 1, 2, and 4.
Which statement best describes one of these transformations?
Triangle 1 is rotated to result in triangle 2.
Triangle 1 is dilated to result in triangle 3.
Triangle 1 is reflected to result in triangle 4.
Triangle 1 is stretched to result in triangle 5.
Answers
A statement which best describes one of these transformations is that: A. triangle 1 is rotated to result in triangle 2.
What are the types of transformation?In Mathematics, there are four (4) main types of transformation and these include the following:
RotationReflectionDilationTranslationWhat is a rotation?In Mathematics, a rotation simply refers to 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.
Since triangle 2 has identical (similar) side lengths and angle measures, but is rotated and triangles 1 and 4 are identical, we can logically deduce that triangle 1 underwent a rotation to produce triangle 2.
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Select all of the following equation(s) that are quadratic in form.
x4 – 6x2 – 27 = 0
3x4 = 2x
2(x + 5)4 + 2x2 + 5 = 0
6(2x + 4)2 = (2x + 4) + 2
6x4 = -x2 + 5
8x4 + 2x2 – 4x = 0
Answers
Answer:
Step-by-step explanation:
x4 – 6x2 – 27 = 0
x=2.49,-2.49
3x4 = 2x
x=6
If a₁ = 6 and an
an-1 + 3 then find the value of a4.
Answers
Answer:
a₄ = 15
Step-by-step explanation:
using the recursive relation [tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] + 3 and a₁ = 6 , then
a₂ = a₁ + 3 = 6 + 3 = 9
a₃ = a₂ + 13= 9 + 3 = 12
a₄ = a₃ + 3 = 12 + 3 = 15
URGENT!! ILL GIVE BRAINLIEST!!!! AND 100 POINTS!!!
Answers
Answer: a is a rational number
Step-by-step explanation
Rational numbers are written in p/q form.
7. Sketch a model to represent the equation
2x + 2 = 12. Then, solve the equation.
Answers
x = 5 will be the correct answer and Model is provided in attachment don't look at the beauty of my figure.
The solution of the equation 2x + 2 = 12 is x = 5. And the model and the graph are given.
What is an equation?Two algebraic expressions having same value and symbol '=' in between are called as an equation.
Given:
Equation 2x + 2 = 12
The model for the equation is given in the attached image.
And the graph is also given in the attached image.
To solve for x:
2x + 2 = 12
Simplifying,
2x = 12 -2
2x = 10
Divide by 2,
2x / 2 = 10 / 2
x = 5.
Therefore, the solution of the equation is x = 5
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-3x-4y=-1, 3x-y=-4 solve by elimination
Answers
Hope it’s correct
y = 1, and x = -1
if you need to show your work:
if you combine the two equations together (eliminating x), it goes like this
-3x - 4y = -1
3x - y = -4
-5y = -5
y = 1
this means:
-3x - 4 = -1
-3x = 3
x= -1
and
3x - 1 = -4
3x = -3
x = -1
(×+13)²=(×+12)²+(×-5)²
Give it to me please :(
Answers
To solve the problem, we will use the law of signs, to solve the problem
Law of signs:- × - = +- × + = -+ × - = -+ × + = +With the law of signs, we solve, but first we must know the following.
¿What are the equations?We know that the equations are those mathematical expressions that are called in members and separated, by their equal sign, which these carry their known data and unknown or unknown data, these are related through their mathematical operations.
Solving problem: x² + 26x + 169 = x² + 24x + 144 + x² - 10x + 25x² + 26x + 169 = 2x² + 14x + 169x² + 26x - 2x² - 14x = 0-x² + 12x = 012x - x² = 0x (12 - x) = 0x = 0.12So, the result of this equation is x = 0.12
¡Hope this helped!
Answer:
x = 0, x = 12
Step-by-step explanation:
Given equation:
[tex](x+13)^2=(x+12)^2+(x-5)^2[/tex]
Expand:
[tex]\implies (x+13)(x+13)=(x+12)(x+12)+(x-5)(x-5)[/tex]
[tex]\implies x^2+26x+169=x^2+24x+144+x^2-10x+25[/tex]
Collect and combine like terms on the right side of the equation:
[tex]\implies x^2+26x+169=x^2+x^2+24x-10x+144+25[/tex]
[tex]\implies x^2+26x+169=2x^2+14x+169[/tex]
Subtract 169 from both sides:
[tex]\implies x^2+26x=2x^2+14x[/tex]
Subtract x² from both sides:
[tex]\implies 26x=x^2+14x[/tex]
Subtract 26x from both sides:
[tex]\implies 0=x^2-12x[/tex]
[tex]\implies x^2-12x=0[/tex]
Factor out the common term x:
[tex]\implies x(x-12)=0[/tex]
Apply the zero-product property:
[tex]\implies x=0[/tex]
[tex]\implies x-12=0 \implies x=12[/tex]
Solution:
x = 0, x = 121. The sum of 4th and 6th progression is 42. The sum of 3rd and 9th term of the progression is 52. Find a). First term b.) the common difference. c). the sum of the first 10 terms of the progression.
Answers
Answer:
a) The first term of the progression is 4.5.
b) The common difference of the progression is 4.
c) The sum of the first 10 terms of the progression is 225.
Step-by-step explanation:
To solve this problem, you can use the formula for the sum of an arithmetic series. An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed number, called the common difference, to the preceding term.
The sum of an arithmetic series with n terms and a common difference d is given by:
Sum = n/2 * (2a + (n-1)d)
Where a is the first term of the series and d is the common difference.
In this problem, you are given that the sum of the 4th and 6th terms is 42 and the sum of the 3rd and 9th terms is 52. You can use these two equations to solve for a and d.
First, let's find the sum of the 4th and 6th terms:
Sum = 4/2 * (2a + 3d) = 42
This simplifies to:
2a + 3d = 21
Next, let's find the sum of the 3rd and 9th terms:
Sum = 6/2 * (2a + 5d) = 52
This simplifies to:
2a + 5d = 26
Now that we have two equations, we can solve for a and d by using substitution or elimination.
To use substitution, we can solve the second equation for d:
d = (26 - 2a)/5
Then, we can substitute this expression for d into the first equation:
2a + 3((26 - 2a)/5) = 21
This simplifies to:
2a + 3(26 - 2a)/5 = 21
Which simplifies to:
2a + 3(26)/5 - 3(2a)/5 = 21
This simplifies to:
2a + 3(26)/5 - 6a/5 = 21
This simplifies to:
-4a + 3(26)/5 = 21
This simplifies to:
-4a + 39 = 21
This simplifies to:
-4a = -18
This simplifies to:
a = 4.5
Now that we know the value of a, we can substitute it back into one of the original equations to find the value of d:
2(4.5) + 3d = 21
This simplifies to:
9 + 3d = 21
This simplifies to:
3d = 12
This simplifies to:
d = 4
Now that we have found the values of a and d, we can use the formula for the sum of an arithmetic series to find the sum of the first 10 terms of the progression:
Sum = 10/2 * (2(4.5) + (10-1)(4))
= 10/2 * (9 + 36)
= 10/2 * 45
= 225
Therefore, the sum of the first 10 terms of the progression is 225.
Answer:
Hence,
Common difference is
First term is
Sum of first 10 terms is 47
Step-by-step explanation:
a4 + a6 =42 eq1
a3 + a9 =52 eq
a1=?
d=?
S10=?
From eq 1,
a4 + a6=42
a1+3d + a1+5d =42
2a1 + 8d= 42 eq 3
From eq 2,
a3 + a9 =52
a1+2d + a1+8d = 52
2a1 + 10d = 52 eq 4
Subtract eq 3 from eq
2a1 + 10d =52
-2a1 -8d = -42
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2d = 10
d=5....
≈=======================
Putting value of d in eq
2a1 +10 d=52
2a1+ 50 =52
a1 =1
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Now we find sum of first 10 terms,
S10 = 2a1 + 9d
S10 = 2+45
S10 = 47
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