A recycling truck that feeds on petroleum is would not be an application of Stuart Wilkinson's theory.
What is about Stuart Wilkinson's theory?A recycling truck that runs on petroleum is designated as option E and would not be a practical application of Stuart Wilkinson's thesis in terms of merging the concept of beneficial robotic capabilities in terms of consuming food.This is referred to as a machine that can automatically mimic some human movements and functions with the help of an external control system.A recycling truck that feeds on petroleum is inaccurate since petroleum isn't a form of food because Stuart Wilkinson's thesis aims to combine the concept of beneficial robotic capabilities in terms of consuming food.To learn more about Stuart Wilkinson's theory refer to:
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a basketball player made 62 out of 80 free throw shots. what percent of the free throws did she make
NO SPAM OR LINKS!!! WILL MARK BRAINLIEST!!!
Review the Monthly Principal & Interest Factor chart to answer the question:
APR 30-Year Term 20-Year Term 15-Year Term
5.5 $5.68 $6.88 $8.17
6.0 $6.00 $7.16 $8.44
6.5 $6.32 $7.46 $8.71
7.0 $6.65 $7.75 $8.99
7.5 $6.99 $8.06 $9.27
The Apolaro household will be financing a principal balance of $115,200.00 for their property. They are trying to decide whether to take a 30-year or 15-year term mortgage. Based on their credit score, they qualify for an APR of 7.0%. Calculate the difference in total interest they would pay between the two terms. (2 points)
$81,012.83
$84,353.72
$89,372.16
$92,416.64
Answer: c
Step-by-step explanation:
Answer:
Step-by-step explanation:
its $89,372 i did the math took the test
Select the figure with an angle of 300°
on a circle in standard position.
(a)
An angle in standard position with terminal side at five-sixths of a full revolution counterclockwise.
(b)
An angle in standard position with terminal side at one-twelfth of a full revolution counterclockwise.
(c)
An angle in standard position with terminal side at one-third of a full revolution counterclockwise.
(d)
An angle in standard position with terminal side at two-thirds of a full revolution counterclockwise.
Note that the figure with an angle of 300° on a circle in standard position is: "An angle in standard position with the terminal side at two-thirds of a full revolution counterclockwise." (Option D)
What is an Angle in Standard Position?A standard position angle is one measured counterclockwise from the positive x-axis in a coordinate plane.
This signifies that the angle's starting side is on the positive x-axis, while the angle's terminal side is in the first or fourth quadrant. The angle's vertex is placed at the origin.
Angles are commonly measured in degrees or radians, with a full rotation being comparable to 360 degrees or 2 radians. Angles may now be readily graphed and compared to one another in a coordinate plane.
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Which of the following arrow diagrams represent a mapping? If that is the case give the domain and range.
The arrow diagram that represents a mapping include the following: arrow diagram C.
The domain of this arrow diagram is equal to {a, c, d}
The range of this arrow diagram is equal to {q, r, s}.
What is a domain?In Mathematics, a domain can be defined as the set of all real numbers for which a particular function is defined.
This ultimately implies that, a domain is the set of all possible input numerical values (x-values) to a function and the domain of a graph comprises all the input numerical values (real numbers) which are located on the x-coordinate.
In this scenario, we can reasonably infer and logically deduce that only the above mentioned relation (arrow diagram C) represents a function because each input variable (domain) has exactly an output variable (range) as illustrated in the diagram.
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Drag the tiles to the correct boxes to complete the pairs.
Match the units of measurement to their International System (SI) equivalents.
1 foot
1 pound
1 fluid ounce
29.5 milliliters-
30 centimeters or 0.30 meters-
450 grams or 0.45 kilograms-
Answer:
see below
Step-by-step explanation:
1 foot = 30 centimeters or 0.30 meters-
1 pound =450 grams or 0.45 kilograms-
1 fluid ounce = 29.5 milliliters-
Please help me on this
Here we are interested in finding the slope of the line. From the given graph we can see that that the line passes through points (6,0) and (0,-8) . We know that slope can be calculated by ,
[tex]\longrightarrow slope \ (m)=\dfrac{y_2-y_1}{x_2-x_1} =\tan\theta [/tex]
[tex]\longrightarrow m =\dfrac{y_2-y_1}{x_2-x_1}\\[/tex]
[tex]\longrightarrow m =\dfrac{0-(-8)}{6-0}\\[/tex]
[tex]\longrightarrow m =\dfrac{8}{6}\\ [/tex]
[tex]\longrightarrow m =\dfrac{4}{3}\\ [/tex]
[tex]\longrightarrow \underline{\underline{ m = 1.\bar{3}}}[/tex]
And we are done!
scenario 3-6 a researcher wishes to study how the average weight y (in kilograms) of children changes during the first year of life. he plots these averages versus the age x (in months) and decides to fit a least-squares regression line to the data with x as the explanatory variable and y as the response variable. he computes the following quantities. r
The slope of the least-squares line is: 0.30
The correct answer is an option (A)
We have following information:
correlation between X and Y (r) = 0.9
mean of the values of X ([tex]\bar{X}[/tex]) = 6.5
mean of the values of Y ([tex]\bar{Y}[/tex]) = 6.6
the standard deviation of the values of X ([tex]S_{X}[/tex]) = 3.6
the standard deviation of the values of Y ([tex]S_{Y}[/tex]) = 1.2
We know that the slope of a least squares regression can be calculated by the formula,
slope m = r [tex]\frac{S_\bar{Y}}{S_\bar{X}}[/tex],
where, r is the correlation coefficient
Let us assume that for given scenario, m represents the slope of of the least-squares line.
Using above formula,
⇒ m = r [tex]\frac{S_\bar{Y}}{S_\bar{X}}[/tex]
⇒ m = 0.9 × [tex]\frac{1.2}{3.6}[/tex]
⇒ m = 0.9 × 0.333
⇒ m = 0.2997
⇒ m ≈ 0.3
Thus, the slope is 0.30
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The complete question is:
A researcher wishes to study how the average weight Y (in kilograms) of children changes during the first year of life. He plots these averages versus the age X (in months) and decides to fit a least- squares regression line to the data with X as the explanatory variable and Y as the response variable. He computes the following quantities.
r = correlation between X and Y = 0.9
X = mean of the values of X = 6.5
Y = mean of the values of Y = 6.6
S_(X) = standard deviation of the values of X = 3.6
S_(Y) = standard deviation of the values of Y = 1.2
The slope of the least-squares line is: ?
A. 0.30 B. 0.88 C. 1.01 D. 2.70 E. 3.00
Francisco added a new sidewalk to his front yard. The
diagram represents Francisco's sidewalk. Determine
the area of the sidewalk to the nearest square foot.
A 10 ft²
B 16.28 ft²
C 3.14 ft²
D 13.14 ft²
Is this correct?
The required of the sidewalk to the nearest square foot is 13.14 ft².
What is area of rectangle?
The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area. In essence, the area of a rectangle is equal to the sum of its length and breadth.
Explain are of circle.A circle's area is the area that it takes up in a two-dimensional plane. It can be simply calculated using the formula A = r2, (Pi r-squared), where r is the circle's radius. The square unit, such as m2, cm2, etc., is the unit of area.
According to question:We have,
Length = 5 ft, width = 2 ft, Radius of circle = 2 ft
Now,
Area of sidewalk = area of rectangle + area quadrant
Area of sidewalk = (5×2) + π(2)^2/4
Area of sidewalk = 10 + 3.14
Area of sidewalk = 13.14 ft²
Thus, required area of the sidewalk is 13.14 ft².
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find the bake if x and y
The value of x is 76°. The solution has been obtained using the properties of hexagon.
What is a hexagon?
A closed, two-dimensional, six-sided polygon in geometry is called a hexagon. It has six vertices, which together create six internal angles, and six line segments. The total of a hexagon's internal angles is 720°.
We know that sum of angles on a straight line is 180°.
So, on the first line with angle 60°, the other angle will be
180° - 60° = 120°
Similarly, on the second line with angle 64°, the other angle will be
180° - 64° = 116°
Similarly, on the third line with angle 36°, the other angle will be
180° - 36° = 144°
Similarly, on the fourth line with angle 48°, the other angle will be
180° - 48° = 132°
Similarly, on the fifth and sixth lines with angle x°, the other angles will be
180° - x° = (180-x)°
We know that sum of angles of interior angles of a hexagon is 720°
Therefore,
⇒ 120° + 116° + 144° + 132° + (180-x)° + (180-x)° = 720°
⇒ 512° + 180° - x° + 180° - x° = 720°
⇒ 872° - 2x° = 720°
⇒ 2x° = 152°
⇒ x = 76°
Hence, the value of x is 76°.
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9b² – 100
Factor it
4 2/3 - 3/4 answer? Please help me anwser this
Answer: 3 11/12
Step-by-step explanation:
Rewrite the equation with the parts seperated:
4 + 2/3 − 3/4
Find the LCD of 2/3 and 3/4
(Which is 12)
Keep going
8/12−9/12= −1/12
Finally, subtract 4 − 1/12 to get 3 11/12
Hope this helps!
12 Juma, Ali and Hassan share the profit of their business in the ratios 3:7:9 respectively. If Juma receives sh. 6000. How much profit did the business yield. (3mks)
If the ratio of profit shares is 3:7:9, then the total profit yield is 38000.
What is ratio?
Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other. Two categories can be used to categorise ratios. Part to whole ratio is one, and part to part ratio is the other. The part-to-part ratio shows the relationship between two separate entities or groupings.
The share of Juma's profit is = 3
The share of Ali's profit is = 7
The share of Hassan's profit is = 9
The ratios can be written as 3x : 7x : 9x
The amount of money Juma received is = 6000
The equation for Juma's profit is -
3x = 6000
x = 6000/3
x = 2000
So, the value for Ali's profit -
7x = 7 × 2000
= 14000
The value for Hassan's profit -
9x = 9 × 2000
= 18000
Now, the total profit is represented by the equation -
= 3x + 7x + 9x
= 6000 + 14000 + 18000
= 38000
Therefore, the total profit is 38000.
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A company is expanding its team at a rate of 10% per year. 17 a) Calculate how many members of staff there will be after 1 year if there are initially 200 members of staff.
Step-by-step explanation:
If a company is expanding its team at a rate of 10% per year, and it currently has 200 members of staff, we can calculate the number of staff after 1 year using the formula:
number of staff after 1 year = 200 + (10/100) * 200
number of staff after 1 year = 200 + 20
number of staff after 1 year = 220
So, there will be 220 members of staff after 1 year if there are initially 200 members of staff.
carrie is flying on two different airlines with a heavy bag both airlines have the same weight limit by charge different fees the first airline charges $25 checked bag fee plus an additional four dollars for every pound that the bag is over the weight limit with the second airline and the price is three dollars per pound over the weight limit and addition to a $28 fee for checking the bag Carrie ended up paying the exact amount and baggage fees at the two airlines how many pounds over the limit was Carrie’s bag how much did she have to pay in fees for each airline
Answer: Let x be the weight of the bag over the limit in pounds.
We know that both airlines have the same weight limit and Carrie paid the same baggage fee at each airline, so we can set up the following equation:
25 + 4x = 28 + 3x
Solving for x:
x = 25
So, Carrie's bag was 25 pounds over the weight limit.
To find the fee for each airline, we can substitute x into the equations for the fees:
Fee for the first airline: 25 + 4(25) = $125
Fee for the second airline: 28 + 3(25) = $103
Therefore, Carrie paid $125 in fees for the first airline and $103 in fees for the second airline.
Step-by-step explanation:
A youth group and their leaders visited a museum
Answer:
What is the question please respond so I may give a complete response
graph these equations? y = 3x - 5, y = 1/2x + 5
The solution of the given system of equations is (4, 7).
What is a linear system of equations?A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
The given system of equations are y=3x-5 -------(I) and y=1/2 x+5 -------(II)
Substitute equation (I) in (II), we get
3x-5=1/2 x+5
3x-1/2 x=5+5
2.5x=10
x=4
Substitute x=4 in equation (I), we get
y=3(4)-5
y=7
So, solution is (4, 7)
Therefore, the solution of the given system of equations is (4, 7).
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PLEASE HELP ME IT’S URGENT
The images of the points after the transformation are (6, 1), (3, 6), (5, 2) and (-8, -4)
How to determine the coordinates of the point?r(x-axis) o T(2, -3) (4, 2)
This implies that we reflect the point (4, 2) across the x-axis and then translate it by (x + 2, y - 3)
Mathematically, this is represented as
Image = (x + 2, 3 - y)
Substitute the known values in the above equation, so, we have the following representation
Image = (4 + 2, 3 - 2)
Evaluate
Image = (6, 1)
T(2, -5) o R 90 CCW (4, 2)
This implies that we translate the point (4, 2) by (x + 2, y - 5) and then rotate it by 90 degrees counterclockwise
Mathematically, this is represented as
Image = (5 - y, x + 2)
Substitute the known values in the above equation, so, we have the following representation
Image = (5 - 2, 4 + 2)
Evaluate
Image = (3, 6)
T(1, 4) o r(y-axis) (4, 2)
This implies that we translate the point (4, 2) by (x + 1, y + 4) and then reflect it across the y-axis
Mathematically, this is represented as
Image = (x + 1, 4 - y)
Substitute the known values in the above equation, so, we have the following representation
Image = (4 + 1, 4 - 2)
Evaluate
Image = (5, 2)
r(y-axis) o T(1, 4) (4, 2)
This implies that we reflect the point (4, 2) across the y-axis and then translate it by (x + 4, y + 2)
Mathematically, this is represented as
Image = (-x - 4, y + 2)
Substitute the known values in the above equation, so, we have the following representation
Image = (- 4 - 4, 2 + 2)
Evaluate
Image = (-8, 4)
Hence, the image is (-8, -4)
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Tim+is+attempting+to+bike+a+12+mike+race+he+knows+at+every+third+of+the+race+he+will+need+to+take+a+break+to+stretch+his+leg+how+many+miles+wo
Answer:
he takes a stretch every 4 miles
Step-by-step explanation: 12/3 = 4
Label The Missing angle measures for this triangle.
The measure of the missing angles are 100° and 30°
What is a triangle?A polygon with three vertices, angles and sides.
Given is a triangle XYW, whose side WZ is extended to the point z,
We know that,
The sum of two opposite interior angles of a triangle is equal to the measure of exterior angle,
∠ 1 + 50° = 150°
∠ 1 = 100°
We know that, sum of the interior angles of a triangle is 180°
∠ 1 + ∠ 2 + 50° = 180°
150°-180° = ∠ 2
∠ 2 = 30°
Hence, the measure of the missing angles are 100° and 30°
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The corner store sells pencils for $0.05, $0.10, and $0.15. List all the ways Kay can spend exactly $0.45 on pencils.
Nine pencils at $0.05 each, 2. Five pencils at $0.10 each and three pencils at $0.05 each 3.Six pencils costing $0.05 apiece and three pencils costing $0.15 each. Pencils: four for $0.15 each, and five for $0.05 each.
What is combination theory?This question uses the combination theory, which states that the total number of combinations of a set of items is equal to the number of variations in one item multiplied by the number of variations in the second item, and so on.
in this question
1. Nine pencils at $0.05 each: Kay can buy nine pencils for $0.45. Nine pencils times $0.05 each equal $0.45.
2. Five pencils at $0.10 each and three pencils at $0.05 each: Kay can buy five pencils for $0.50 and three pencils for $0.15.
Five pencils times $0.10 each equal $0.50 and three pencils times $0.05 each equal $0.15.
Adding the two totals together equals $0.45.
3. Three pencils at $0.15 each and six pencils at $0.05 each: Kay can buy three pencils for $0.45 and six pencils for $0.30.
Three pencils times $0.15 each equals $0.45 and six pencils times $0.05 each equal $0.30.
Adding the two totals together equals $0.45.
4. Four pencils at $0.15 each and five pencils at $0.05 each:
Kay can buy four pencils for $0.60 and five pencils for $0.25.
Therefore, Four pencils times $0.15 each equals $0.60 and five pencils times $0.05 each equal $0.25. Subtracting the two totals together equals $0.45.
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Humber Practice Skills - Estimating REVISION AVAILABLE 2.29 + 9.11 15.381-5.414 7.631.11 + 5.99 5.734 + 6.893 18.355-4.411 Answer the following Estimate the sum or difference of the following decimal numbers by rounding to the nearest tenth. ? CALCULATOR AMI RIGHT E Criteria O Search hp Learning 33 NCFE M DON'T KNOW RESET ROU 11:21 24/01/2023
The solution to the expression with decimals rounded to the nearest tenth is: -2
Examine the hundredth number to round the decimal number to the nearest tenth. If the number is more than 5, multiply it by one. If it is less than 5, leave the tenth place value alone and eliminate any numbers after the tenth. 4.624, for example, is 4.6 rounded to the closest tenth.
Thus, given the expression:
2.29 + 9.11 + 15.381 - 5.414 - 7.631.11 + 5.99 - 5.734 + 6.893 - 18.355 -4.411
We first round them up to the nearest tenths is:
2.1 + 9.1 + 15.4 - 5.4 - 7.6 + 6.0 - 5.7 + 6.9 - 18.4 - 4.4 = -2
Thus, it is correct to state that the version of the above-given expression resolved to the nearest tenths (2.1 + 9.1 + 15.4 - 5.4 - 7.6 + 6.0 - 5.7 + 6.9 - 18.4 - 4.4) = -2
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Julio is buying a car. He makes a down payment of 20%
and he makes arrangements to finance the remaining
$14,000
What is the dollar amount of Julio's down payment?
Julio's down payment would be 20% of the total cost of the car. If he is financing $14,000 of the remaining cost, the total cost of the car would be $14,000 / (1 - 0.20) = $14,000 / 0.80 = $17,500. Therefore, Julio's down payment would be $17,500 * 0.20 = $3,500.
What is a percentage in plain English?
A percentage is a portion of a total stated as a number between 0 and 100 as opposed to being expressed as a fraction. The percentages of everything are 100%, nothing is 0%, 50% of something is 50%, and nothing is 0%. To calculate the percentage, divide the component by the total and multiply the result by 100.
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Using percentage, Julio's down payment would be $3,500.
What does a % mean in everyday terms?In contrast to being expressed as a fraction, a percentage is a portion of a total expressed as a number between 0 and 100. The percentages are as follows: 100% of everything, 0% of nothing, 50% of something, and 0% of nothing. Divide the component by the total, then multiply the result by 100 to get the percentage.
Julio's down payment would be 20% of the total cost of the car.
If he is financing $14,000 of the remaining cost,
the total cost of the car would be
$14,000 / (1 - 0.20)
= $14,000 / 0.80
= $17,500.
Therefore, Julio's down payment would be $17,500 * 0.20 = $3,500.
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for each limit in exercise 55-64 use graphs and algebra to approximate the largest delta or the smallest magnitude n taht corresponds to the given value of epsilon or m, according to the appropriate formal limit definition
The largest δ value of the given limit [tex]\lim_{x \to 1^+} \frac{1}{x^2-1}=\infty[/tex] and M=1000 is calculated. The required δ value is [tex]\delta=\sqrt{1+10^{-3}}-1[/tex].
When no values are defined, a limit is used to assign values to certain functions. It is possible to assess the limits using algebraic techniques. A collection of guidelines for manipulating limits when using other operators is known as the algebra of limits. Given the expression [tex]\lim_{x \to 1^+} \frac{1}{x^2-1}=\infty[/tex] and M=1000. When x>1, we write this expression as,
[tex]\begin{aligned} \frac{1}{x^2-1}& > 1000\\x^2-1& < 1000^{-1}=10^{-3}\\x^2& < 10^{-3}+1\\x& < \sqrt{1+10^{-3}}\end{aligned}[/tex]
Now, choosing, [tex]\delta=\sqrt{1+10^{-3}}-1[/tex] when we have δ > 0. Therefore, [tex]\sqrt{1+10^{-3}} > 1[/tex]. Then, [tex]x\in(1, 1+\delta)[/tex]. This is given by,
[tex]\begin{array}{ccc}1& < x& < 1+\delta\\1& < x& < \sqrt{1+10^{-3}}\end{array}[/tex]
This implies, [tex]\frac{1}{x^2-1} > 1000=M[/tex].
Therefore, the largest δ value will be [tex]\sqrt{1+10^{-3}}-1[/tex].
The complete question is -
For each limit, use graphs and algebra to approximate the largest δ or the smallest magnitude N that corresponds to the given value of ∈ or M, according to the appropriate formal limit definition.
[tex]\lim_{x\to 1^+} \frac{1}{x^2-1}=\infty[/tex], M=1000, find largest δ > 0.
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Sophie is a dog that loves to play catch. Unfortunately, she isn't very good, and the probability that she catches a ball is only 0.2. Let x be the number of tosses required until Sophie catches a ball. (a) Does x have a binomial or a geometric distribution? (b) What is the probability that it will take exactly two tosses for Sophie to catch a ball? P(exactly two tosses) (c) What is the probability that more than three tosses will be required? P(more than three tosses)
X has a geometric distribution because it is the number of trials (tosses) required until a success (catching the ball) occurs.
b) The probability that it will take exactly two tosses for Sophie to catch a ball is P(x=2) = (0.2)(0.8)^1 = 0.16. This is calculated by multiplying the probability of success (0.2) by the probability of failure (0.8) raised to the power of the number of trials (2-1=1)
c) To find the probability that more than three tosses will be required, we need to find the probability that Sophie does not catch the ball on the first three tosses and then catch it on the fourth toss.
This can be found by summing the probabilities of Sophie not catching the ball on the first three tosses (0.8^3 = 0.512) and Sophie catching the ball on the fourth toss (0.8^3*0.2 = 0.1024). The probability that more than three tosses will be required is P(x>3) = 1 - P(x≤3) = 1 - (0.512 + 0.1024) = 0.3856.
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What is the slope of the following linear graph 2/3 -2/3 3/2 -3/2
The slope of following points (3, 3) and (6,1) is -2/3 .
What is Linear equation ?
Linear equation can be defined as the equation in which the highest degree is one.
Given points are (3, 3) and (6,1)
we know that slope = y2-y1 / x2-x1
y2-y1 = 1-3
The difference of y -intercept is -2
= -2
x2-x1 = 6-(3)
= 6-3
The difference of x-intercept is 3
= 3
So, slope could be y2-y1 / x2-x1 that is -2/3.
Therefore, The slope of following points (3, 3) and (6,1) is -2/3 .
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14. Calculate the x-intercepts of the equation by factoring:
8x²-16x = 0
A liter of petrol cost 93 for an equation to find the cost in pound of any number of liter of petrol
The equation for finding cost y for x liters of petrol is y = 93x
What is an equation?An equation is a mathematical statement that shows that two mathematical expressions are equal.
Given that, A liter of petrol cost £93,
Since, the unit rate of the petrol is £93 per liter
If we say let x be the number of liters we need, the total cost will have to pay will be = 93x
And, if total cost is y, then,
y = 93x
Hence, the equation for finding cost y for x liters of petrol is y = 93x
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#20 i need help with this question
Answer:
Step-by-step explanation:
tha answer is 69
What is the solution to this equation: 42 +82=²? (Round answer to the nearest tenth.)
O c= 8.9
O e = 12
Oc=6.3
O c=3.5
The solution to this equation: 42 +82=x² is x = 12.
What is square number?A square number, also known as a perfect square, is an integer that is the square of another integer; in other words, it is the result of multiplying another integer by itself. For instance, 9 is a square number because it equals 32 and can be written as 3 3.
Instead of the product n n, the equivalent exponentiation n2, which is typically pronounced "n squared," is used to represent a number's square. The name of the shape is where the term "square number" originated. The area of a unit square (1 × 1) is the unit of area.
The solution to this equation: 42 +82=x²
x = √(42 +82)
x = √(144)
x = 12
Thus, The solution to this equation: 42 +82=x² is x = 12.
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NO LINKS!!
There is a mistake in each of these problems. Please help
Answer:
ii) Reduce :
[tex]$\begin{aligned} 5 \ln x-2 \ln (x y) & =\ln x^5-\ln (x y)^2 \\ & =\ln \left(\frac{x^5}{(x y)^2}\right) \\ & =\ln \left(\frac{x^5}{x^2 y^2}\right) \\ & =\ln \left(\frac{x^3}{y^2}\right)\end{aligned}$[/tex]
iii ) Expand :
[tex]\begin{aligned}\log _3\left(\frac{x^5}{y}\right)^4 & =4 \log _3\left(\frac{x^5}{y}\right) \\& =4 \log _3 x^5-4 \log _3 y \\& =20 \log _3 x-4 \log _3 y\end{aligned}[/tex]
Answer:
[tex]\textsf{ii)} \quad \ln \left(\dfrac{x^3}{y^2}\right)[/tex]
[tex]\textsf{iii)} \quad 20 \log_3 x-4 \log_3y[/tex]
Step-by-step explanation:
Question iiGiven expression:
[tex]5 \ln x-2 \ln (xy)[/tex]
[tex]\textsf{Apply the power law}: \quad n \ln x=\ln x^n[/tex]
[tex]\implies \ln (x^5) - \ln ((xy)^2)[/tex]
[tex]\textsf{Apply the quotient law}: \quad \ln x - \ln y=\ln \left(\dfrac{x}{y}\right)[/tex]
[tex]\implies \ln \left(\dfrac{x^5}{(xy)^2}\right)[/tex]
This is where the error occurred in the original calculation as the log quotient law was applied incorrectly.
[tex]\textsf{Apply exponent rule} \quad (ab)^c=a^{c}b^c\quad \textsf{to the denominator}:[/tex]
[tex]\implies \ln \left(\dfrac{x^5}{x^2y^2}\right)[/tex]
Simplify the fraction:
[tex]\implies \ln \left(\dfrac{x^3}{y^2}\right)[/tex]
Question iiiGiven expression:
[tex]\log_3\left(\dfrac{x^5}{y}\right)^4[/tex]
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies 4\log_3\left(\dfrac{x^5}{y}\right)[/tex]
[tex]\textsf{Apply the log quotient law}: \quad n\log_a \left(\dfrac{x}{y}\right)=n\log_ax - n\log_ay[/tex]
[tex]\implies 4 \log_3 x^5-4 \log_3y[/tex]
This is where the error occurred in the original calculation as the coefficient 4 was not applied to the both logarithms when applying the log quotient law.
[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]
[tex]\implies 5 \cdot 4 \log_3 x-4 \log_3y[/tex]
[tex]\implies 20 \log_3 x-4 \log_3y[/tex]