The least distance from point P to line l is approximately equal to 3.536 units.
How to find the least distance between a line and a point outside the line
There are several ways to calculate the least distance, in this case we decided to use the approach based on vectors and projections. First, we generate two auxiliary vectors:
Vector parallel to the line l
[tex]\overrightarrow {AB} = (5, 1) - (2, 4)[/tex]
[tex]\overrightarrow{AB} = (3, - 3)[/tex]
A vector from one point on the line l to the point P
[tex]\overrightarrow{AP} = (1, 1) - (2, 4)[/tex]
[tex]\overrightarrow {AP} = (- 1, - 3)[/tex]
Now we proceed to find the vector distance by using the following vectorial formula:
[tex]\vec d = \frac{\overrightarrow{AP} \,\bullet\, \overrightarrow {AB}}{\overrightarrow {AB} \,\bullet \,\overrightarrow {AB}} \cdot \overrightarrow{AB} - \overrightarrow {AP}[/tex]
[tex]\vec d = \frac{(-1)\cdot (3) + (- 3) \cdot (- 3)}{3^{2}+ (-3)^{2}} \cdot (3, - 3) - (-1, - 3)[/tex]
[tex]\vec d = - \frac{1}{6}\cdot (3, - 3) - (- 1, - 3)[/tex]
[tex]\vec d = \left(-\frac{1}{2}, \frac{1}{2} \right) + (1, 3)[/tex]
[tex]\vec d = \left(\frac{1}{2}, \frac{7}{2} \right)[/tex]
The magnitude of the vector distance is determined by Pythagorean theorem:
[tex]d = \sqrt{\left(\frac{1}{2} \right)^{2}+\left(\frac{7}{2} \right)^{2}}[/tex]
d ≈ 3.536
The least distance from point P to line l is approximately equal to 3.536 units.
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prove that sin4a+2sin2acos2a+cos4a=1
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
[tex] \qquad❖ \: \sf \: \sin {}^{4} ( \alpha) + 2 \sin {}^{2} ( \alpha) \cos {}^{2} ( \alpha) + { \cos {}^{4} ( \alpha) }^{} [/tex]
[tex] \qquad❖ \: \sf \:( \sin {}^{2} ( \alpha) ) {}^{2} + 2( \sin {}^{2} ( \alpha))( \cos {}^{2} ( \alpha)) + {( \cos {}^{2} ( \alpha)) }^{} [/tex]
( use identity a² + 2ab + b² = (a + b)² )
[tex] \qquad❖ \: \sf \:( {\sin {}^{2}( { \alpha}) + \cos {}^{2} ( \alpha)) }^{2} [/tex]
( sin² x + cos ² x = 1 )
[tex] \qquad❖ \: \sf \: {1}^{2} [/tex]
[tex] \qquad❖ \: \sf \: {1}^{} [/tex]
[tex] \qquad \large \sf {Conclusion} : [/tex]
Value of that expression is 1
Solve the problems using your graphing calculator. Round all parameter values to
the nearest hundredth. Complete Steps 1-4 in order.
Caleb is training for an upcoming race. He runs a mile every day, keeps track of his
timing, and records his progress.
Day #
1
2
3
4
5
6
7
Run time (min)
8.2
8.1
7.5
7.8
7.4
7.2
7.1
Step 1: Choose the function that best represents the data.
y= 8.36x -0.19
y=-0.19x+8.36
y = 8.36x+0.19
y=0.19x8.36.
Drawing a scatter plot for the given values on a graphing calculator, we have;
y = -0.19•x + 8.36How can the function that best represents the data be found?The values are;
Day; Running time
1; 8.2
2; 8.1
3; 7.5
4; 7.8
5; 7.4
6; 7.2
7; 7.1
Entering the above values in a graphing calculator, we have;
The slope of the line of best fit = -0.19The x-intercept = 45Therefore;
0 = -0.19 × 45 + y-intercept0 = -8.36 + y-intercept
y-intercept = 8.36The equation found using the statistics function for scatter plot on a calculator is therefore;
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help! I need help, pls show work!
Let us first define what any of these terms mean.
The mean is found by adding all the numbers in the set, and dividing the sum by how many numbers are found in the set.
> Given set {5, 12, 6, 3, 8, 16, 8, 6}, the sum of the numbers is 5 + 12 + 6 + 3 + 8 + 16 + 8 + 6 = 64. 64 ÷ 2 = 32. Thus, the mean is 32.
The median is found by finding the number in the middle of the set, and dividing it by 2. This is easy in odd numbers as there is only one number in the middle that is the same amount of numbers left and right proportionally. However, in even numbers, there are two middles to be proportional. If it as an even number, then add the 2 middles together, and divide by 2.
> Given set {5, 12, 6, 3, 8, 16, 8, 6}, this set has an even number of numbers in the set. Thus, the middles are 3 and 8. 3 + 8 = 11, and 11 ÷ 2 = 5.5. Thus, the median is 5.5.
The mode is the number that most frequently appears in the set. In this case, we have 2 frequent numbers. That is 6 and 8. They both appear 2 times separately. Thus, 6 and 8 are the modes of this set.
The range is the difference (i.e subtraction) between the largest number on the set and the smallest number on the set.
> Given set {5, 12, 6, 3, 8, 16, 8, 6}, the range is 16-3 or 13. Thus, the range is 13.
Now that we have the definitions, we can compute the other set very easily.
> Given set {2, 7, 4, 11, 12, 4, 6}, the mean is 18.
> Given set {2, 7, 4, 11, 12, 4, 6}, the median is 5.5.
> Given set {2, 7, 4, 11, 12, 4, 6}, the mode is 4.
> Given set {2, 7, 4, 11, 12, 4, 6}, the range is 10.
Hope it helped!
15. When x = -3, what is the value of
|4x²| − 2x − (9 − |−3x|)
(A)0
(B) 12
(℃) 30
(D) 42
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Solving for your equation:}[/tex]
[tex]\mathbf{|4x^2| - 3x - (9 - |-3x|)}[/tex]
[tex]\mathbf{= |(4)(9)|-(2)(-3)-(9-|(-3)(-3)|)}[/tex]
[tex]\mathsf{= |36| - (2)(-3)- (9 - |(-3)(-3)|)}[/tex]
[tex]\mathbf{=36 - (2)(-3) - (9 - |(-3)(-3)|)}[/tex]
[tex]\mathbf{= 36 -(-6) - (9 - |(-3)(-3)|)}[/tex]
[tex]\mathbf{= 42 - (9 - |(-3)(-3)|)}[/tex]
[tex]\mathbf{= 42 - (9 - |9|)}[/tex]
[tex]\mathbf{= 42 - (9 - 9)}[/tex]
[tex]\mathbf{= 42 - (0)}[/tex]
[tex]\mathbf{= 42 - 0}[/tex]
[tex]\mathbf{= 42}[/tex]
[tex]\huge\textbf{Notice that your equation kept getting}\\\huge\textbf{getting smaller and smaller?}\\\\\\\huge\textbf{That's because we kept solving as we got}\\\huge\textbf{closer to your result and we got there}\\\huge\textbf{sooner than we thought :)}[/tex]
[tex]\huge\textsf{ANYWAY!}[/tex]
[tex]\huge\textsf{Therefore, the answer is: \boxed{\mathsf{Option\ D. 42}}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
SOMEONE PLS HELP, ITS GEOMETRY, ITS URGENT ASAP PLS
Answer:
for st1: It's Given
For St2: Definition
For St3: Dividing BE as BF and FE or You can write From Figure:
For St4: You can write From st3
for st5: From St 4
For St7 : They are congruent by SSS axiom
can someone please help mee ( will give brainliest 20 points!!!)
Answer:
1. can't answer, can only approximate
2. when f(x) = 2, x = -1
Step-by-step explanation:
We can't evaluate f(2) because we don't have f(x), if we did, we would plug in 2 for x.
On the other hand, solve f(x) = 2 asks for what x is equal to when f(x) = 2, f(x) is just the value on the y axis in relationship with x (in short terms, the value on the y axis).
So looking at the graph, we cam see that when x = -1, f(x) = 2.
If m=5 and n = 2, find the value of 3mn
Answer:
30
Step-by-step explanation:
3mn=3x5x2=30
Identify the range of the function shown in the graph.
Answer:
D. 0≤y≤5
Step-by-step explanation:
The range refers to how far on the y axis a function exists on a graph. This function starts at y=0 and ends at y=5, meaning you can find a point on the function for any y-level between 0 and 5. So y is greater than or equal to 0, and is less than or equal to 5.
A: A function with a range of all real numbers goes both up and down infinitely, however this function does not go up and down infinitely so we can rule out A.
B: If y>0, the function would start above y-level 0 and go up infinitely, however this graph does not display that so we can rule out B as well.
C: C is the domain of the graph, which is how far left and right the function exists.
Someone gotta help me out, it’s geometry and I’m in a hurry and I can’t figure this out. Someone help pls :(
Equilateral Triangle: All 3 sides of a Triangle are equal in length
Isosceles Triangle: Two sides of Triangle are equal in length
Scalene: Triangle has different side lengths
So, for example, when you see Q1, you estimate the length of each side and think it is Scalene Triangle because all of its sides are of different length. (You can also use a ruler to find out the side lengths if it can be used for scale)
Q2, you can say that the side lengths AC and BC look similar, so it is an Isosceles Triangle
Q6, when you look at the side lengths, you can tell it is an equilateral Triangle because all the sides are equal in length
You can try the rest and check for yourself. If you still have any doubts, you could ask me. I hope this information helps you.
What is the equation for the cones volume?
Answer: D
Step-by-step explanation:
[tex]V=\frac{1}{3}\pi r^{2} h[/tex] is the general formula for the volume of a cone with radius r and height h.
Substituting in s for both r and h, [tex]V=\frac{1}{3}\pi s^{3}[/tex].
The__________or_______a value of the function is found at the vortex.
Answer:
Step-by-step explanation:
n fluid dynamics, a vortex is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil. Vortices are a major component of turbulent flow. The distribution of velocity, vorticity, as well as the concept of circulation are used to characterise vortices. Hope that gave you a clue and hope you learned something new.
The minimum or maximum value of a function is found at the vertex.
The vertex represents either the minimum or maximum point on the graph of the function.
If the quadratic function opens upward, the vertex is the minimum point, and if it opens downward, the vertex is the maximum point.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic function in the form ax² + bx + c.
Once you find the x-coordinate, you can substitute it back into the function to obtain the corresponding y-coordinate, which represents the minimum or maximum value of the function at the vertex.
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Bobby is a freshman in high school and he just received $500 for graduation from middle school from his grandparents. he wants to save his $500 for college in 4 years if he deposits his $500 into a simple interest savings account for the next 4 years that has a rate of 2.85%. how much will he have in his account after 4 years?
Bobby will have $557 in his account after 4 years.
What is simple interest?Simple interest is calculated based on a loan's principal or the initial deposit into a savings account.
Simple interest doesn't compound, therefore a creditor will only charge interest on the principal sum, and a borrower will never be required to pay further interest on the interest that has already accrued.
Simple interest = (principal amount×Rate of interest×time-period)/100
The principal amount deposited was $500.
The rate of interest given by the bank is 2.85%.
The duration of deposit is 4 years.
Simple interest = (500×2.85×4)/100
= 57
Simple interest given by the bank after 4 years is $57.
The total sum of money Bobby will have in his bank is $500 + $57 = $557.
Therefore, after 4 years Bobby will have $557 in his account.
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If DAB = CBA,
DAB = 35° and CBA = 8x + 3
X= ?
Answer:
35= 8x + 3
35 - 3 = 8x
32 = 8x
32/8= 8x/8
x = 4
If g stands for a whole number, find the first three possible values of g2 + 3.
3, 4, 7
1, 4, 7
3, 5, 7
0, 4, 7
The first three values of g² + 3 are 3, 4, 7. The correct option is the first option 3, 4, 7
Evaluating an expressionFrom the question, we are to determine the first three possible values of g² + 3
From the given information,
g stands for a whole number
∴ The first three values of g are 0, 1, 2
When g = 0
g² + 3 becomes
0² + 3
= 0 + 3
= 3
When g = 1
g² + 3 becomes
1² + 3
= 1 + 3
= 4
When g = 2
g² + 3 becomes
2² + 3
= 4 + 3
= 7
Hence, the first three values of g² + 3 are 3, 4, 7. The correct option is the first option 3, 4, 7.
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Find (in radians) for an
angle having an arc length of
23 and a radius of 15.
?
0 = ² radians
Enter
By using the concept of circumference and all known inputs, we conclude that an arc length of 23 and a radius of 15 is equal to 23/15 radians.
How to determine the central angle of an arc
According to the statement, arc length (s) and radius (r) are given:
s = 23, r = 15
The central angle (θ), in radians, is determine by dividing the former by the latter:
θ = s/r
θ = 23/15 rad
By using the concept of circumference and all known inputs, we conclude that an arc length of 23 and a radius of 15 is equal to 23/15 radians.
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A swimming pool is being filled with water. The diameter of the pool is 27 feet, and the height is 4 feet. Which formula best represents the volume of the pool?
V = pi (13.5) (4)
V = pi (13.5) squared (4)
V = pi (27) (4)
V = pi (27) squared (4)
The expression that best represents the volume is (b) [tex]V = \pi (13.5)^2 (4)[/tex]
How to determine the volume?The given parameters are:
Diameter, d = 27
Height, h = 4
Start by calculating the radius using:
r = d/2
This gives
r = 27/2
r = 13.5
The volume is then calculated as:
[tex]V = \pi r^2 h[/tex]
This gives
[tex]V = \pi (13.5)^2 (4)[/tex]
Hence, the expression that best represents the volume is (b) [tex]V = \pi (13.5)^2 (4)[/tex]
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Noise levels at 5 airports were measured in decibels yielding the following data:
152,154,139,124,120
1. Construct the 80% confidence interval for the mean noise level at such locations. Assume the population is approximately normal.
2. Calculate the sample mean for the given sample data. Round your answer to one decimal place.
The confidence interval for the mean noise at locations is (127.09,148.51).
Given noise levels at 5 airports=152,154,139,124,120
We have to find the confidence interval for the mean noise and sample mean of the data. We know that mean is the sum divided by the number of items taken.
Mean = sum of X values/n
=137.8 (calculated in figure)
s=[tex]\sqrt{(x-x bar)^{2} /n-1}[/tex]
=[tex]\sqrt{972.8/4}[/tex]
=[tex]\sqrt{243.2}[/tex]
=15.59
t value at 80% confidence level with degree of freedom 4 (5-1) is 1.533.
standard error=t value*s/[tex]\sqrt{n}[/tex]
=1.533*15.59/[tex]\sqrt{5}[/tex]
=10.71
Upper level= Mean + standard error
=137.8+10.71
=148.51
Lower level=Mean - standard error
=137.8-10.71
=127.09
Hence the confidence interval for the true mean noise is (127.09,148.51) and sample mean is 137.8.
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The formula used to convert degrees Celsius to degrees Fahrenheit is
F = 0+32.
Convert 59°F to degrees Celsius. Solve the formula for C, and then use it to
convert the temperature.
Which is the correct formula and conversion?
A. C= F-32; conversion: 59°F = 135°C
B. C= (F-32); conversion: 59°F = 15°C
C. C= (F-32); conversion: 59°F = 106°C
OD. C=F-32; conversion: 59°F = 15°C
Answer:
See below
Step-by-step explanation:
Wrong formula ...and all of your answers are messed up too....
F = 9/5 C + 32
re-arrange to:
C = 5/9 ( F-32)
C = 5/9 (59-32) = 15 ° C
What is the shaded portion of the figure?
Answer:
D
Step-by-step explanation:
Area of Part of a Circle = (θ/360)[tex]\pi r^{2}[/tex]
We first find the area of CAB first.
Radius of CAB = AB = 6cm
θ = 112
Area of CAB = [tex]\frac{112}{360} \pi (6)^{2} \\=\frac{56}{5} \pi cm^{2}[/tex]
Next we will find the area of FAE.
Radius of FAE = AE = 4cm
Area of FAE = [tex]\frac{112}{360} \pi (4)^{2} \\=\frac{224}{45} \pi cm^{2}[/tex]
Lastly,
Area of Shaded Region = Area of CAB - Area of FAE
= [tex]\frac{56}{5} \pi -\frac{224}{45}\pi \\=\frac{56}{9} \pi cm^{2}[/tex]
Therefore the answer is D.
The following expression gives an approximate value of the total average credit card debt in a u.s. household (in dollars) t years after 1995. 404t+5190 use this expression to predict what the total average credit card debt was or will be in the year 1998. answer: in the year 1998, the total average credit card debt for a u.s. household will be (or was) dollars.
The total average credit card debt for a U.S household will be $9292.
How much is considered a lot of credit card debt?If your total balance is more than 30% of the total credit limit, you may be in too much debt. Some experts consider it best to keep credit utilization between 1% and 10%, while anything between 11% and 30% is typically considered good.
We are given that an expression which gives an approximate value of the total average credit card debt in a U.S. household (in dollars )t years after 1995
We have to find the total average credit card debt will be in the year 2004.
Number of years =2004-1995=9
t=9 years
Substitute the value of t in the given expression
Then , total average credit card debt in a U.S household will be in the year
Debt(t)=418t+5530
Debt(9)=418 * 9+5530= $9292 in 2004
Hence, in the year 2004, the total average credit card debt for a U.S household will be $9292.
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which pair of anglers are alternate exterior angles of parallel lines v and w
In the image given, a pair of alternate exterior angles is: a° and d°.
What are Alternate Exterior Angles?The exterior angles that lie opposite sides along the transversal that intersects two parallel lines are said to be alternate exterior angles, which are congruent.
In the image given below, angles a and d lie alternate to each other along the transversal and are also exterior angles.
Therefore, a pair of alternate exterior angles is: a° and d°.
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Given ƒ(x) = x + 10 and g(x) = x2 + 5x - 1, find ƒ(3) + g(5).
A sales tax of 7% was added to the $36.00 price of a jacket. What was the total cost of the jacket?
a. $36.70
b. $38.10
c. $36.07
d. $38.52
Answer:
[tex]\huge\boxed{\sf \$ \ 38.52}[/tex]
Step-by-step explanation:
Marked price = $36.00
Sales tax = 7%
Sales tax in $:= 7% of marked price
= 7% of $36 [Key: "of" means "to multiply"]
= 7/100 * 36
= 0.07 * 36
= $2.52
Selling price:= Marked price + sales tax
= $36 + $2.52
= $38.52
[tex]\rule[225]{225}{2}[/tex]
The total cost of the jacket was the amount of $38.52. The correct answer is option (d).
To find the total cost of the jacket after adding a 7% sales tax, we need to calculate the tax amount and add it to the original price.
Calculate the tax amount:
Tax amount = 7% of $36.00
Tax amount = 0.07 × $36.00
Tax amount = $2.52
Add the tax amount to the original price:
Total cost = $36.00 + $2.52
Total cost = $38.52
Thus, the total cost of the jacket was the amount of $38.52.
Hence, the correct answer is option (d).
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Given that
5x:9=8:3
Calculate the value of x
Give your answer in its simplest form.
Answer:
24/5
Step-by-step explanation:
5x : 9
(8 : 3)×3
5x : 9
24 : 9
5x = 24
x = 24/5 = 4/4/5
Hello!
5x : 9 = 8 : 3
5x × 3 = 8 × 9
15x = 72
5x = 24
x = 24/5
x = 4.8
∴ The value of x is 4.8.
Jun Wei is a bus driver who earns $ 1600 per month while Lixin is a manager who earns $ 6800 per
month. In 2012, Jun Wei donated a total of $ 1200 to charitable organisations while Lixin donated a
total of $4500 to charitable organisations. Who donated a higher percentage of his or her annual
income to charitable organisations?
Answer:
Jun Wei donated a higher percentage than Lixin
calculate it by multiplying monthly salary by 12 then divide the total donated to total earned for both people... you Will Find 6.25 % for Jun Wei but for Lixin You Will get 5.51%
i hope this helps u :)
thank u
Select the correct answer.
Simplify the following radical expression.
√48
16√3
4√3
4√2
6√2
Step-by-step explanation:
[tex] = \sqrt{48} [/tex]
[tex] = \sqrt{16 \times 3} [/tex]
[tex] = \sqrt{ {4}^{2} \times 3 } [/tex]
[tex] = \sqrt{ {4}^{2} } \times \sqrt{3} [/tex]
[tex] = 4 \times \sqrt{3} [/tex]
[tex] = 4 \sqrt{3} [/tex]
The answer is B.
What is the probability that a standard number cube will show a two or four when it is rolled?
Choose the term that best fills in the blank to complete the statement.
A dilation is a transformation that maps every point on a line segment to a point on a parallel line segment. The image has a length that is determined by the preimage and a scale factor, which is a fixed _[blank]_ of the original segment's length.
a. Distance
b. Length
c. Multiple
d. Width
Answer:
d
Step-by-step explanation:
becoz u already answered it lloll
Which statement best defines perpendicular lines?
A.
lines that share a point
B.
lines that intersect and form right angles
C.
lines that lie in the same plane and do not intersect
D.
lines that lie in the same plane
Reset Next
Perpendicular lines are lines that intersect and form right angles
What is a line?A line is defined as the shortest distance between two points. Its equation is written in slope-intercept form as y = mx + b
Perpendicular lines on the other hand are lines that meet at a point to form a right angle. Hence we can conclude that perpendicular lines are lines that intersect and form right angles
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Step-by-step explanation:
2x - 15 ≤ 5 pls help
[tex]2x - 15 \leqslant 5 \\ 2x \leqslant 20 \\ x \leqslant 10[/tex]
Interval Notation: x ε ( -∞ , 10 ][tex]\Huge\boxed{x\leq10}[/tex]
To solve for [tex]x[/tex], you need to isolate it on one side of the equation.
Start by adding [tex]15[/tex] to both sides.
[tex]\begin{aligned}2x-15+15&\leq5+15\\2x&\leq20\end{aligned}[/tex]
Now, divide both sides by [tex]2[/tex] for the final answer.
[tex]\begin{aligned}\frac{2x}{2}&\leq\frac{20}{2}\\x&\leq10\end{aligned}[/tex]