B. 18
Explanation
The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number
it is given by.
[tex]\begin{gathered} u=(u_x,u_y) \\ v=(v_{_x},v_y) \\ u\cdot v=(u_xv_x+u_yv_y) \end{gathered}[/tex]so, we can find the dot product by multiplying the corresponding values in each vector and adding them together
Step 1
get the dot product
let
[tex]\begin{gathered} v_1=(2,4) \\ v_2=(-1,5) \end{gathered}[/tex]then
[tex]\begin{gathered} v_1\cdot v_2=(2\cdot-1)+(4\cdot5) \\ v_1\cdot v_2=-2+20 \\ v_1\cdot v_2=18 \end{gathered}[/tex]therefore, the answer is
B. 18
I hope this helps you
What is the equation of the parabola shown below, given a focus at F(1, 5) and a directrix of x = −3? In addition, identify the vertex and the equation of the axis of symmetry for the parabola.
EXPLANATION
First, let's find the vertex.
From the graph, the vertex is (-1, 5).
It is symmetric about y = 5
Length of the Latus rectom (a) =2 x 4 = 8
Therefore, the equation of the graph is;
[tex]y=\frac{1}{a}(y-5)^2-1[/tex]Substitute a = 8
[tex]x=\frac{1}{8}(y-5)^2-1[/tex]a right cone has a radius of 27 cm and a height of 36cm. find the slant height of the cone. find the surface area of the cone. find the volume of the cone
A right triangle is formed where the radius is one leg, the height is the other leg and the slant height is the hypotenuse. Applying the Pythagorean theorem:
[tex]\begin{gathered} c^2=a^2+b^2 \\ s^2=27^2+36^2 \\ s^2=729+1296 \\ s^2=2025 \\ s=\sqrt[]{2025} \\ s=45\operatorname{cm} \end{gathered}[/tex]The surface area of a right cone is calculated as follows:
[tex]SA=\pi rs+\pi r^2[/tex]where r is the radius and s is the slant height. Substituting with r = 27 cm and s = 45 cm, we get:
[tex]\begin{gathered} SA=\pi\cdot27\cdot45+\pi\cdot27^2 \\ SA=1215\pi+729\pi \\ SA=1215\pi+729\pi \\ SA=1944\pi\approx6107.25\operatorname{cm} \end{gathered}[/tex]The volume of a right cone is calculated as follows:
[tex]V=\pi\cdot r^2\cdot\frac{h}{3}[/tex]where h is the height. Substituting with r = 27 cm and h = 36 cm, we get:
[tex]\begin{gathered} V=\pi\cdot27^2\cdot\frac{36}{3} \\ V=\pi\cdot729\cdot12 \\ V=8748\pi\approx27482.65\operatorname{cm}^3 \end{gathered}[/tex]In February of 2014, gas was about $3.37 per gallon. In February 2015, gas was about 2.25 per gallon. What is the percent decrease from 2014 to 2015?
According to the problem, the price decrease from $3.37 to $2.25. Let's find the difference
[tex]3.37-2.25=1.12[/tex]Then, we divide
[tex]\frac{1.12}{3.37}=0.33[/tex]Hence, the answer is 33%.A recent study of 28 city residents showed that the mean of the time they had lived at their present address was 9.3 years. The standard deviation of the population was 2 years. Find the 90% confidence interval of the true mean? Assume that the variable is approximately normally distributed. Show all your stepsVery confused in this exercise I’m self teaching myself
Given that:
- The sample size is 28 city residents:
[tex]n=28[/tex]- The mean of the time (in years) they had lived at their present address was:
[tex]\mu=9.3[/tex]- The standard deviation (in years) of the population was:
[tex]\sigma=2[/tex]Then, you need to use the following formula for calculating the Confidence Interval given the Mean:
[tex]C.I.=\mu\pm z\frac{\sigma}{\sqrt{n}}[/tex]Where μ is the sample mean, σ is the standard deviation, "z" is the z-score, and "n" is the sample size.
By definition, the z-score for a 90% confidence interval is:
[tex]z=1.645[/tex]Therefore, you can substitute values into the formula and evaluate:
[tex]C.I.=9.3\pm(1.645)(\frac{2}{\sqrt{28}})[/tex]You get that the lowest value is:
[tex]9.3-(1.645)(\frac{2}{\sqrt{28}})\approx8.678[/tex]And the highest value is:
[tex]9.3+(1.645)(\frac{2}{\sqrt{28}})\approx9.922[/tex]Hence, the answer is:
[tex]From\text{ }8.678\text{ }to\text{ }9.922[/tex]Find the volume of the cylinder. Round your answer to the nearest tenth. Use 3.14 for a.4 ft17 ftThe volume of the cylinder is about|ft3
we know that
The volume of the cylinder is equal to
[tex]V=B\cdot h[/tex]where
B is the area of the base and h is the height of cylinder
we have that
[tex]B=\pi\cdot r^2[/tex]we have
pi=3.14
r=4 ft
substitute
[tex]\begin{gathered} B=3.14\cdot4^2 \\ B=50.24\text{ ft\textasciicircum{}2} \end{gathered}[/tex]Find the volume
[tex]V=B\cdot h[/tex]we have
B=50.24 ft^2
h=17 ft
substitute
[tex]\begin{gathered} V=50.24\cdot17 \\ V=854.1\text{ ft\textasciicircum{}3} \end{gathered}[/tex]the volume is about 854.1 cubic feetI need help with math I have dyscalculia and I don’t understand
The summation symbol means that we need to add the terms (n-1) for n = 1, 2, 3, and 4.
In other words, we have the following terms:
[tex]\begin{gathered} 1-1=0 \\ \\ 2-1=1 \\ \\ 3-1=2 \\ \\ 4-1=3 \end{gathered}[/tex]And we need to add them:
[tex]0+1+2+3=6[/tex]Therefore, the result is 6.
In the regular octagon below, if AP = 10 cm. and BC = 15 cm, find it's area.
Given:
There is a regular octagon given as below
Required:
We want to find the area of given regular octagon if AP = 10 cm. and BC = 15 cm
Explanation:
As we can see in the figure that there 8 triangles which are exactly same
so if we find area of 1 triangle and multiply with 8 we get the area of whole regular octagon
The area of 1 triangle is
[tex]a=\frac{1}{2}*10*15=75\text{ cm}^2[/tex]now to find area of regular octagon
[tex]A=8a=75*8=600\text{ cm}^2[/tex]
Final answer:
600 sq cm
Point M is the midpoint of AB. The coordinates of point A are (-7,2) and the coordinates of M are (-3, 1).What are the coordinates of point B?The coordinates of point B are
We have three points that lie on a line namely, A M B, in a cartesian coordinate system.
The coordinates of each point are as such:
[tex]A\text{ ( -7 , 2 )}[/tex]The mid-point of a line segment ( AB ) is given by coordinate ( M ) as follows:
[tex]M\text{ ( -3 , 1 )}[/tex]We will define the coordinates of point ( B ) in terms of cartesian coordinates as follows:
[tex]B\text{ ( x , y ) }[/tex]Here the line ( A B ) with points [ A, M , B ] are colinear. This means they all lie on the same line passing through all three points.
A straight line always have a constant slope/gradient ( m ) . This slope is determined between two points that lie on the line.
The formulation of calculating the slope ( m ) of the line in a cartesian coordinate system is as follows:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]The respective cartesian coordinates of each of the two points are represented by sub-scripts of ( x and y ).
Here we will take two points ( A and M ) to determine the slope ( m ) of the line passing throgh all three points as follows:
[tex]\begin{gathered} m\text{ = }\frac{1\text{ - 2}}{-3\text{ -(-7)}}\text{ = }\frac{1-2}{-3+7} \\ m\text{ = }\frac{-1}{4} \end{gathered}[/tex]This slope ( m ) holds true for the entire line and must satisfy all consecutive points that lie on the line.
We will now consider points ( A and B ) and determine the slope ( m ):
[tex]\begin{gathered} m\text{ = }\frac{y\text{ - 2}}{x\text{ - (-7)}} \\ \\ -\frac{1}{4}\text{ = }\frac{y\text{ - 2}}{x+7} \\ \\ -\text{ ( x + 7 ) = 4}\cdot(y-2) \\ -x\text{ - 7 = 4y - 8} \\ \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ = 1 - 4y }}\textcolor{#FF7968}{\ldots}\text{\textcolor{#FF7968}{ Eq 1}} \end{gathered}[/tex]What we did above was calculated the slope ( m ) between two points ( A and B ) which resulted in an expression in terms of coordinates of point B ( x and y ). We equated that expression with the value of slope ( m ) that holds true for all points that lie on the line.
Then we used the slope equation and expressed the x-coordinate of point B in terms of y-coordinate of point B. This relaitonship is termed as Equation 1 ( Eq 1 ).
Next, we were also given that point M is the midpoint of line segment ( AB ). Using the definition of a mid-point ( M ) i.e the magnitude of line segments are:
[tex]|AM|\text{ = |MB|}[/tex]The magnitudes of line segment AM and MB must be equal for point ( M ) to be a mid-point of line segment ( AB ).
We will express the formulation of determining a magnitude of line segment using two points:
[tex]\text{length of line segment = }\sqrt{(\times_2-x_1)^2+(y_2-y_1)2}[/tex]So for equating the lengths (magnitudes) of line segments ( AM ) and ( MB ) we have:
[tex]\begin{gathered} A\text{ ( -7 , 2 ) , M ( -3 , 1 ) , B ( x , y )} \\ \sqrt{(-3-(-7))^2+(1-2)^2}\text{ = }\sqrt{(-3-x)^2+(1-y)^2} \end{gathered}[/tex]Square both sides of the equation:
[tex](4)^2+(-1)^2=(-1)^2\cdot(x+3)^2+(1-y)^2[/tex]Evaluate left hand side of equation and apply PEMDAS at right hnd side of the equation:
[tex]\begin{gathered} 16+1=(x^2+6x+9)+(y^2\text{ - 2y + 1 )} \\ 17\text{ = }x^2+y^2\text{ + 6x - 2y + 10} \\ \textcolor{#FF7968}{7}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{x^2+y^2}\text{\textcolor{#FF7968}{ + 6x - 2y }}\textcolor{#FF7968}{\ldots Eq2} \end{gathered}[/tex]We have two equations with two unknows ( x and y ) as follows:
[tex]\begin{gathered} x\text{ = 1 - 4y }\ldots Eq1 \\ 7=x^2+y^2\text{ + 6x - 2y }\ldots Eq2 \end{gathered}[/tex]We will solve the two equation by simultaneous substitution method. Substitute Eq1 into Eq2 as follows:
[tex]\begin{gathered} 7=(1-4y)^2+y^2\text{ + 6}\cdot(1\text{ - 4y ) - 2y} \\ 7=(1-8y+16y^2)+y^2\text{ + ( 6 - 24y ) - 2y} \\ 7=17y^2\text{ -34y + 7 } \\ 0=17y^2\text{ -34y} \\ 0\text{ = y}\cdot(17y\text{ - 34 )} \\ \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = 0 OR y = }}\textcolor{#FF7968}{\frac{34}{17}}\text{\textcolor{#FF7968}{ = 2}} \end{gathered}[/tex]We will now plug the two values of coordinate ( y ) into ( Eq 1 ) and solve for ( x ):
[tex]\begin{gathered} y\text{ = 0, x = 1 - 4}\cdot(0)\text{ = 1} \\ y\text{ = 2, x = 1 -4}\cdot(2)=-7\text{ } \end{gathered}[/tex]We have two solutions for the coordinates of point ( B ) as follows:
[tex]B\text{ : ( 1 , 0 ) OR ( -7 , 2 )}[/tex]However, point B must have only one pair of coordinate. So we have to investigate both solutions given above and reject a redundant solution.
We see that solution B: ( -7 , 2 ) is redundant solution, hence, rejected. This is because it represents the coordinates of point A: ( -7 , 2 ) - given in question. So two different points can not attain the same set of coordinates! Hence,
The solution to the set of coordinates of point B is:
[tex]\textcolor{#FF7968}{B\colon}\text{\textcolor{#FF7968}{ ( 1 , 0 ) }}[/tex]
You flip a coin 100 times and record each outcome.You find that you land on heads 47 times and land ontails 53 times. What is the theoretical probability oflanding on heads?
If we were asked the "experimental probability", the answer would have been
[tex]\frac{47}{100}\text{ that is number of outcomes/total outcomes. }[/tex]But we were asked the "theoretical probability". And in theory, we can get either a head or a tail if a coin is flipped. So that makes it
[tex]\frac{1\text{ outcome }}{total\text{ outcomes of 2 }}\text{ = }\frac{1}{2}[/tex]Therefore, the answer is
[tex]\frac{1}{2}[/tex]May I please get help with finding weather they are quadrilaterals, parallelograms, or rectangles
Given:
Three diagrams are given.
Diagram VWXY:
Quadrilateral, Parallelogram,Rectangle
Diagram IJGH:
Quadrilateral
Diagram ABCD:
Quadrilateral , Parallelogram
The length of a rectangle is four times the width. If the area of the rectangle is 196square inches, then find the length and width
Let the length of the rectangle be "l" and width be "w".
The length is 4 times the width, we can write:
[tex]l=4w[/tex]The area is given as 196. We know area of a rectangle is A = lw, where l is length and w is width. Thus, we can write:
[tex]196=lw[/tex]Substituting the first equation into the second, we can solve for w. The process is shown below:
[tex]\begin{gathered} 196=lw \\ 196=(4w)w \\ 196=4w^2 \\ w^2=\frac{196}{4} \\ w^2=49 \\ w=\sqrt[]{49} \\ w=7 \end{gathered}[/tex]Now, using the 1st equation, we can solve for the length, "l". Shown below:
[tex]\begin{gathered} l=4w \\ l=4\times7 \\ l=28 \end{gathered}[/tex]Answer:
Length = 28 inches
Width = 7 inches
seven more than the product of 22 and a number
Answer:
22n + 7
Step-by-step explanation:
We can let n represent the number being multiplied by 22 (products imply multiplication).
We put the seven after the multiplication since "seven more" means that we're adding the 7 to the product.
Linear functions f(x) = x and g(x) = 8/9x are graphed on the same coordinate plane. Which statement about therelationship between these two graphs is true?
Firstly, let us proceed to plot the graph of f(x) and g(x).
From the graph;
f(x)=x (green line)
g(x)=8/9 x (Blue)
The relationship they have is that they both starts from the origin (0,0).
Also, They both have a positive but not the same slope.
f(x) slope = 1
g(x) slope = 8/9
Point 0 is the center of the circle, What is the value of X
Answer: x = 22
Explanation:
From the information given. O is the center of the circle. The distance from the center of the circle to the circumference is the radius. This means that
OQ = radius
O to the vertex where angle 56 is formed is also a radius
This means that two sides the triangle formed insides the circle are equal. This also means that this triangle is an isosceles triangle. The base angles of an isosceles triangle are equal. This means that
angle 56 = angle Q because they are the base angles of the isosceles triangle
Recall, the sum of the angles in a triangle is 180 degrees. This means that
angle O + 56 + 56 = 180
angle O + 112 = 180
angle O = 180 - 112
angle O = 68
Recall, the angle formed by a tangent with the radius of the circle is 90 degrees. This means that angle Q = 90 degrees
Considering triangle POQ,
angle P + angle O + angle Q = 180
x + 68 + 90 = 180
x + 158 = 180
x = 180 - 158
x = 22
f(x)=(0.13x⁴+0.22x³)-0.88x²-0.25x-0.09for this polynomial use a graph to state the number of turning points
The graph of the function is:
From this, we can conclude that the polynomial has three turning points
Graph the line parallel to x= -1 that passes through (8,4).could you also draw a picture
The line x = -1 is a vertical line, since it has a specific x-coordinate and no y-coordinate.
A line parallel to a vertical line is also a vetical line, so our line will have the equation x = b, where we need to find the value of b.
Since our line passes through the point (8, 4), we know that its x-coordinate will be 8, so our line is x = 8.
Drawing the lines (x = -1 in blue and x = 8 in green), we have:
The answer that should be graphed is just the green line.
use the circle graph to answer the following questionhow many more pop/rock records than soul records were sold in the year shown?
We are given the record sales of varous types.
67 million records were sold.
We are asked to find out how many more Pop/Rock records than Soul records were sold in the year?
From the given information we see that
Pop/Rock records = 56%
Let's find out 56% of 67 million
[tex]67\times\frac{56}{100}=37.52\: \text{million}[/tex]Sour records = 17%
Let's find out 17% of 67 million
[tex]67\times\frac{17}{100}=11.39\: \text{million}[/tex]Now find the difference between Pop/Rock records and Soul records
[tex]difference=37.52-11.39=26.1\: \text{million}[/tex]Therefore, about 26.1 million more Pop/Rock rethan Soul records were sold in the year
find first four terms of an arithmetic series if last term is 10 times first term and sum to n terms is 121
Answer: 120
Step-by-step explanation:
S = n/2 (a(1) + a(n)), where n is the number of terms (10), a(1) is the first term (3), and a(n) is the last term (21).
By substitution, we have,
S = 10/2 (3 + 21)
S = 120
Mrs.Gall orders 240 folders and divides them equally among 3 classes. How many folders does each class receive? What basic fact did you use?
Answer:
80
Step-by-step explanation:
240 folders divided by 3 class
If mQR = 80° and mQS = 150°, what is m
we have that
m
by exterior angle
so
substitute given values
m
mcan Someone help me i still cant understand ,number 3
3)yes, the lines are parallel
Explanation
[tex]\begin{gathered} y=2x+5 \\ y=2x \end{gathered}[/tex]To see whether or not two lines are parallel, we must compare their slopes. Two lines are parallel if and only if their slopes are equal
Step 1
check the slopes
when you have a equation of a line in the form
[tex]\begin{gathered} y=mx+b \\ it\text{ is called, slope intercetp form} \\ \text{where} \\ m\text{ is the slope} \\ \text{and b is the y intercept} \end{gathered}[/tex]hence, for line 1
[tex]\begin{gathered} y=2x+5\rightarrow y=mx+b \\ so \\ m_1=2 \\ b=5 \end{gathered}[/tex]now, for line 2
[tex]\begin{gathered} y=2x\rightarrow y=mx+b \\ so \\ m_2=2 \\ b_2=0 \end{gathered}[/tex]Finally, compare the slopes
[tex]\begin{gathered} m_1=2 \\ m_2=2 \end{gathered}[/tex]the slopes are equal so the lines are parallel
I hope this helps you
2(4+-8)⁶+3 evaluate the Expression
The given expression is
2(4+-8)⁶+3
The first step is to evaluate the bracket.
4 + - 8 = 4 - 8 = - 4
The expression becomes
2(-4)^6 + 3
= 2(4096) + 3
= 8192 + 3
= 8195
A. Determine and then compare the rate of change (slope) for each function in terms of the quantities compared.b. Determine and the compare the y-intercept of each function in terms of the quantities.
A.
Pavilion's line:
[tex]y=10x+50[/tex]The slope of the line is the coefficient multiplying x, so it is m=10
Heliophobia's line:
To determine the slope of this line you have to use the following formula:
[tex]m_p=\frac{y_1-y_2}{x_1-x_2}[/tex]Where
(x₁,y₁) are the coordinates of one point on the line
(x₂,y₂) are the coordinates of a second point on the line
I'll choose to use points (1,85) and (0,70) but you can use any pair of points on the given line:
[tex]m_h=\frac{85-70}{1-0}=\frac{15}{1}=15[/tex]The slope of the Pavillion's line is m=10 → it indicates that y increases 10 units for every unit increase of x.
The slope of the Heliophobia's line is m=15 → it indicates that y increases 15 units for every unit increase of x.
The increase of Heliophobia's line is greater than the increase of Pavillion's line.
B.
The y-intercept is the value of y when x=0
For Pavillion's line the y-intercept is
[tex]\begin{gathered} y=10\cdot0+50 \\ y=50 \end{gathered}[/tex]The coordinates are (0, 50)
For the Heliophobia's line the y-intercept is given in the table (0,70)
Peter works at a sports store. On a particular day, he surveyed a random sample of 50 customers and observed that 30 customers liked to play tennis and 15 customers liked to play basketball.Which of the statements are likely true? Select all that are true.A. 30% of the equipment in the store should be related to basketball.B. 60% of the equipment in the store should be related to tennis.C. 30% of the sales at the sports store that day were related to basketball.D. 70% of the sales at the sports store that day were related to basketball.E. 40% of the sales at the sports store that day were related to tennis equipment. F. 60% of the sales at the sports store that day were related to tennis equipment.
Given data:
Total sample = 50
customers that liked tennis = 30
customers that liked basketball = 15
To select the answers that are true, we can check all the options
Step 1: Find the percentage of those who like tennis and basketball
[tex]\begin{gathered} \text{ The percentage of customers that like tennis is given by} \\ \frac{30}{50}\text{ x 100\% = 60\%} \end{gathered}[/tex][tex]\begin{gathered} \text{The percentage of customers that like basketball is given by} \\ \frac{15}{50}\text{ x 100\% = 30\%} \end{gathered}[/tex]Option A
Since 30% like basketball, then it will be reasonable to have 30% of the equipment related to basketball.
OptionB
Since 60% like tennis, then it will be reasonable to have 60% of the equipment related to tennis
Option C
Since 30% like basketball, then 30% of the sales is likely to be related to basketball
Option D
Since 30% like basketball, then IT IS NOT LIKELY to have 70% of the sales related to basketball.
Option E
Since 60% like tennis, then IT IS NOT LIKELY to have 40% of the sales related to tennis
Option F
Since 60% like tennis, then IT IS LIKELY to have 60% of the sales related to tennis
Hence, we select options
A, B, C, and F as True
Theirs 165 freshman145 Sophomores 114 Juniors102 SeniorsIf a studemt is chosen at random from this group whats the probability a senior will be chosen
To find the probability of choosing a senior student, we have to divide the total number of seniors by the total number of students.
According to the problem, there are 102 seniors. Let's sum to find the total number of students.
[tex]165+145+114+102=526[/tex]Then, we divide these numbers.
[tex]P_{\text{senior}}=\frac{102}{526}=\frac{51}{263}\approx0.19[/tex]The probability of choosing a senior is 51/263 or 19%, approximately.8 divided by 856 long division
Answer: look at the attachment bellow
Find the difference: 75.12 - 2.1 O A. 7.302O B. 73.02O C. 75.11 O D. 54.12
Then, the answer is number B.
157 - 95x + 72 + 13x =
given equation
157-95x+72+13x
First arrange the variables terms together and constant terms together,
157+72-95x+13x
Now simplify constant terms together and variable terms together
229-82x
82x=229
x=229/82
x=2.79
I need to make sure this is correct please graph.
We have the expression:
[tex]y=\frac{4}{5}x+8[/tex]In order to plot the function we replace two values for x and we will get two values for y [Respectively], that is:
x = 0 => y =(4/5)(0)+8 => y = 8
x = 1 => y = (4/5)(1)+8 => y = 8.8
We then have the two points:
(0, 8)
(1, 8.8)
By looking at the fucntion we can tell is a function that describes a line, now we graph:
what is the value of 6 × 7 – 3^2 × 9 + 4^3
Starting with the given expression:
[tex]6\cdot7-3^2\cdot9+4^3[/tex]Follow the hierarchy of operations to find the value.
First, solve potencies and roots. Solve the powers 3^2 and 4^3:
[tex]6\cdot7-3^2\cdot9+4^3=6\cdot7-9\cdot9+64[/tex]Next, solve the multiplications and divisions:
[tex]6\cdot7-9\cdot9+64=42-81+64[/tex]Solve the sums and substractions:
[tex]42-81+64=-39+64=25[/tex]Therefore:
[tex]6\cdot7-3^2\cdot9+4^3=25[/tex]