Answers
Answer:
about 37
Step-by-step explanation:
(8x-23) ----> divide
---- ----
8. 8. ------> 8x cancels out and is just x
x- 2.875+34 =. 37
Related Questions
Sample proportion of .14 and standard deviation of.02, use empirical rule to construct a 95% confidence interval
Answers
The empirical rule states that 65% of the data under the normal curve is within 1 standard deviation of the mean, 95% of the data is within 2 standard deviations of the mean, and 99% is within 3 standard deviations of the mean.
The approximation to the distribution of the sample proportion has the following shape:
[tex]\hat{p}\approx(p;\frac{p(1-p)}{n})[/tex]The mean of the distribution is the sample proportion: μ= p
The standard deviation of the distribution is the square root of the variance
σ=√[p(1-p)/n]
For the given distribution:
μ= 0.14
σ= 0.02
95% of the distribution is μ ± 2σ
Upper bound:
[tex]\mu+2\sigma=0.14+2\cdot0.02=0.18[/tex]Lower bound:
[tex]\mu-2\sigma=0.14-2\cdot0.02=0.10[/tex]The 95% confidence interval is [0.10;0.18]
A bag of popcorn contains 36 ounces. Your friend ate 14 of the bag. You eat 1/3 of what's left. How many ounces of popcorn did you eat?
Answers
Answer:
you ate 7.33333 (repeating) ounces of popcorn
Step-by-step explanation:
36 - 14 = 22
22/3 = 7.3333 (repeating)
What answer shows two pairs of adjacent angles in the figure?angles 2 and 5; angles 2 and 4angles 1 and 4; angles 2 and 5angles 4 and 5; angles 2 and 4angles 1 and 4; angles 1 and 2
Answers
Step 1
Given;
[tex]A\text{ pair of angles in a figure}[/tex]Required; To find which answer shows two pairs of adjacent angles in the figure.
Step 2
Adjacent angles are two angles that have a common side and a common vertex (corner point) but do not overlap in any way.
Therefore, the pair of adjacent angles is;
[tex]\text{Angles 4 and 5; Angles 2 and 4}[/tex]Hence, the answer is: angles 4 and 5; angles 2 and 4
Sphere 6.2 in diameter
Answers
ANSWER:
128.84 cubic inches.
STEP-BY-STEP EXPLANATION:
We have that the volume of a sphere is given by the following equation:
[tex]V=\frac{4}{3}\cdot\pi\cdot r^3[/tex]The radius equals half the diameter, therefore we can calculate the radius as follows:
[tex]\begin{gathered} r=\frac{d}{2}=\frac{6.2}{2} \\ r=3.1 \\ \text{replacing and we calculate the volume:} \\ V=\frac{4}{3}\cdot\frac{22}{7}\cdot3.1^3 \\ V=128.84 \end{gathered}[/tex]The volume of the sphere with the diameter of 6.2 is 128.84 cubic inches.
Factor the polynomial: s^2+ 12s + 32
Answers
SOLUTION
We want to factor the polynomial
[tex]s^2+12s+32[/tex]To do this we look for two values with s such that when we multiply them, we get 32 and when we add then we get the middle item 12s.
These are 8s and 4s because
[tex]\begin{gathered} 8s+4s=12s \\ 8s\times4s=32s^2 \end{gathered}[/tex]Now we replace 8s and 4s with the middle item and factorize, we have
[tex]\begin{gathered} s^2+12s+32 \\ s^2+8s+4s+32 \\ s(s+8)+4(s+8) \\ (s+4)(s+8) \end{gathered}[/tex]Hence the answer is
(s + 4) (s + 8)
complete the flow proof . complete parts a. through d.
Answers
KS is common in both the triangles
so, to complete SSS rule KS = KS will be the answer.
so the answer is b
The properties of integer addition and subtraction also apply topolynomial
addition and subtraction.
•_____under additionand subtraction
•Commutative property of addition
•____property of addition
Answers
Polynomial is commutative under addition and non-commutative under subtraction.
Given, the properties of integer addition and subtraction.
Now, we have to check the properties apply to polynomial or not
Let, a polynomial be p(x) = 2x²+3x+5
and other polynomial be q(x) = x²+2x+3
Now, on adding, we get
p(x) + q(x)
(2x²+3x+5) + (x²+2x+3) = 3x²+5x+8
also, q(x) + p(x)
(x²+2x+3) + (2x²+3x+5) = 3x²+5x+8
So, as p(x) + q(x) = q(x) + p(x)
Therefore, polynomial is commutative under addition
Now, on subtracting, we get
p(x) - q(x)
(2x²+3x+5) - (x²+2x+3) = x²+x+2
also, q(x) - p(x)
(x²+2x+3) - (2x²+3x+5) = -x²-x-2
So, as p(x) + q(x) ≠ q(x) + p(x)
Therefore, polynomial is non-commutative under subtraction.
Hence, polynomial is commutative under addition and non-commutative under subtraction.
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Solve the system using graphing, substitution or elimination. If needed round soulutions to the nearest tenth
Answers
Given
The system of equations,
[tex]\begin{gathered} 9x+y=45\text{ \_\_\_\_\_\lparen1\rparen} \\ x^3-3x^2-25x+93=y\text{ \_\_\_\_\_\_\lparen2\rparen} \end{gathered}[/tex]To find the solution.
Explanation:
It is given that,
[tex]\begin{gathered} 9x+y=45\text{ \_\_\_\_\_\lparen1\rparen} \\ x^3-3x^2-25x+93=y\text{ \_\_\_\_\_\_\lparen2\rparen} \end{gathered}[/tex]From (1),
[tex]y=45-9x[/tex]Substitute y in (2).
Then,
[tex]\begin{gathered} x^3-3x^2-25x+93=45-9x \\ x^3-3x^2-25x+9x+93-45=0 \\ x^3-3x^2-16x+48=0 \\ x^2(x-3)-16(x-3)=0 \\ (x-3)(x^2-16)=0 \\ (x-3)(x^2-4^2)=0 \\ (x-3)(x-4)(x+4)=0 \end{gathered}[/tex]That implies,
[tex]\begin{gathered} x-3=0,x-4=0,x+4=0 \\ \text{ }x=3,\text{ }x=4,\text{ }x=-4 \end{gathered}[/tex]Therefore, for x=3,
[tex]\begin{gathered} y=45-9\times3 \\ =45-27 \\ =18 \end{gathered}[/tex]For x=4,
[tex]\begin{gathered} y=45-9\times4 \\ =45-36 \\ =9 \end{gathered}[/tex]For x=-4,
[tex]\begin{gathered} y=45-(9\times-4) \\ =45+36 \\ =81 \end{gathered}[/tex]Hence, the solution set is (3,18), (4,9), (-4,81).
suppose that your boss must choose for employees in your office to attend a conference in Jamaica. Because all 14 of you want to go . he decides that the only fair way is to draw names out of a hat. What is the probability that you, Samuel, Emma and Paul are chosen? Into a fraction or round your answer to four decimal places if necessary
Answers
suppose that your boss must choose four employees in your office to attend a conference in Jamaica. Because all 14 of you want to go . he decides that the only fair way is to draw names out of a hat. What is the probability that you, Samuel, Emma and Paul are chosen? Into a fraction or round your answer to four decimal places if necessary
total employees=14
probability that you are chosen is P=1/14
probability that Samuel are chosen is P=1/13
the probability that Emma are chosen is P=1/12
the probability that Paul are chosen is P=1/11
therefore
the probability that you, Samuel, Emma and Paul are chosen is
P=(1/14)(1/13)(1/12)(1/11)=4.16x10^-5=0.00004
problem N 2
P=(1/27)(1/26)(1/25)=0.00005698=0.0001
1) What is the remainder when 3x3 - 4x2 - 14x + 3 is divided by3x+5?A)A.43B)wiu0WI)D)IM
Answers
SOLUTION
The given polynomail is
[tex]3x^3-4x^2-14x+3[/tex]To be divided by
[tex]3x+5[/tex]Since the question requires to find the remainder
Then following remainder theorem
Set 3x+5 to zero and solve for x
[tex]\begin{gathered} 3x+5=0 \\ x=-\frac{5}{3} \end{gathered}[/tex]Substitute x=-5/3 into the given polynomial to get the remainder
[tex]\begin{gathered} 3(-\frac{5}{3})^3-4(-\frac{5}{3})^2-14(-\frac{5}{3})+3 \\ =3(-\frac{125}{27})-4(\frac{25}{9})+14(\frac{5}{3})+3 \\ =-\frac{125}{9}-\frac{100}{9}+\frac{70}{3}+3 \\ =\frac{-125-100+210+27}{9} \\ =\frac{12}{9} \\ =\frac{4}{3} \end{gathered}[/tex]Therefore, the remainder is
[tex]\frac{4}{3}[/tex]Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second day, 4 cents on the third day and so on.
Hint: use an= a1 (r)^n-1 and Sn= a1 (1-r^n) / 1 - r
A. If the daily wage keeps doubling, what would your income be on day 31? Give your answer in dollars NOT pennies.
Income on day 31 = $ __________
B. If the daily wage keeps doubling, what will your total income be for working 31 days? Give your answer in dollars NOT pennies.
Total Income for working 31 days = $ _________
Answers
The amount earned, calculated using the formula for geometric progressions are as follows;
(A) The income on day 31 =$1,073,741,824
(B) Total income for 31 working days = $2,147,483,647
What is a geometric progression?A geometric progression is one in which each subsequent term is a constant multiple of the previous term
(A) The function that indicates the amount earned on the nth day is the formula for a geometric progression , which can be presented as follows;
[tex]a_n = a_1\cdot r^{n-1}[/tex]
The sum is presented as follows;
[tex]S_n =a_1\cdot \dfrac{1 - r^n}{1 - r}[/tex]
Therefore;
The common ratio, r = 2
The first term, a₁ = 1
The amount received on day 31 is therefore;
[tex]a_{31} = 1\times 2^{31-1} = 1,073,741,824[/tex]
Income on day 31 = $1,073,741,824
(B) The total income in 31 days is therefore;
[tex]S_{31} =a_1\cdot \dfrac{2^{31} - 1}{2- 1}== 2,147,483,647[/tex]
The total income in 31 working days = $2,147,183,647
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AC, DF, and GI are parallel. Use the figure to complete the proportion.JFFE?DE
Answers
Given that AC, DF, and GI are parallel, we can see that line JH bisects angle J. This means that triangles formed with the parallel lines are similar. Considering triangle JDF,
JF corresponds to JD
FE corresponds to DE
Thus, the ratios are
JF/FE = JD/DE
Find the equation of the circle that has a diameter with endpoints located at (7,3) and (7,-5). A. (x-7)²+(x+ 1)² = 16 B. (x-7)² + (-1)² =4 C. (x+1)²+(y-7)= 16 D. (x-7)²+(y+ 1)²= 64
Answers
Answer:
The equation of the circle is;
[tex]\mleft(x-7\mright)^2+\mleft(y+1\mright)^2=16[/tex]Explanation:
Given that the circle has a diameter with endpoints located at (7,3) and (7,-5).
The diameter of the circle is the distance between the two points;
[tex]\begin{gathered} d=\sqrt[]{(7-7)^2+(3--5)^2_{}} \\ d=\sqrt[]{(0)^2+(3+5)^2_{}} \\ d=\sqrt[]{64} \\ d=8 \end{gathered}[/tex]The radius of the circle is;
[tex]\begin{gathered} r=\frac{d}{2}=\frac{8}{2} \\ r=4 \end{gathered}[/tex]The center of the circle is at the midpoint of the line of the diameter.
[tex]\begin{gathered} (h,k)=(\frac{7+7}{2},\frac{3-5}{2}) \\ (h,k)=(\frac{14}{2},\frac{-2}{2}) \\ (h,k)=(7,-1) \end{gathered}[/tex]Applying the equation of a circle;
[tex](x-h)^2+(y-k)^2=r^2[/tex]Substituting the given values;
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ (x-7)^2+(y+1)^2=4^2 \\ (x-7)^2+(y+1)^2=16^{} \end{gathered}[/tex]Therefore, the equation of the circle is;
[tex]\mleft(x-7\mright)^2+\mleft(y+1\mright)^2=16[/tex]What is -1.47 rounded to four decimal places as needed?
Answers
Given the negative number
[tex]-1.47[/tex]Four decimal places implies there should be four digits after the decimal point. in the absence of no digit, add zero to complete the number of required decimal place. To round the number above to four decimal places, we will have
[tex]-1.4700[/tex]Hence, -1.47 rounded to four decimal places is -1.4700
How to convert 9.4 degrees into feet and inches
Answers
The value of 9.4 degrees in feet is 0.1884 feet and in inches is 2.26 inches.
To calculate feet from degrees, divide the angle by 360, multiply by 2 times pi, then finally, multiply by the radius.
And feet to inches we know 1 foot equals 12 inches.
Converting degrees to feet is commonly discussed while looking at a circle and converting an angle to the length of the arc of that angle. When discussing latitude and longitude, that circle is frequently the earth.
The following formula is used to convert an angle in degrees to length in feet.
L = (a/360) * 2 * pi * r
Where,
L is the length in the feet.
a is the angle in degrees.
r is the radius of the circle in feet.
We are given 9.4 degrees.
Let r = 1 feet
Put the given values in the above formula, we will get;
L = (9.4/360) * 2 * pi * 1
L = 0.03 * 2 * 3.14 * 1
L = 0.1884 feet
L = 0.1884 * 12 = 2.26 inches.
Thus, the value of 9.4 degrees in feet is 0.1884 feet and in inches is 2.26 inches.
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hi! how do i find surface area of a cylinder? for some reason i can’t get the right answer
Answers
Use the formula above to find the surface area of a cylinder.
For the given cylinder the given data is the height (6mi) and diameter (16mi), use the diameter to find the radius:
[tex]\begin{gathered} r=\frac{d}{2} \\ \\ r=\frac{16mi}{2}=8mi \end{gathered}[/tex]Surface area:
[tex]\begin{gathered} SA=2\pi(8mi)\placeholder{⬚}^2+2\pi(8mi)(6mi) \\ SA=128\pi mi^2+96\pi mi^2 \\ SA=224\pi mi^2 \end{gathered}[/tex]Then, the surface area of the given cylinder is: 224π square milesPls help I attached the question
Answers
Answer:
C
Step-by-step explanation:
A Nintendo Switch handheld game has a screen that is 4 inches tall and 9.4 inches long, what is the diagonal distance across the screen? A. 13.4 inchesB. 10.22 inchesC. 9.4 inchesD. None of the above
Answers
EXPLANATION:
To calculate the diagonal distance we must follow the following steps:
-The length corresponds to the horizontal dimension of the screen.
-The height corresponds to the vertical dimension of the screen.
Now we must apply Pythagoras' theorem
[tex]\begin{gathered} a^2+b^2=c^2 \\ a=4 \\ b=9.4 \\ c=diagonal\text{ distance} \\ a^2+b^2=c^2 \\ (4)^2+(9.4)^2=c^2 \\ 16+88.36=c^2 \\ 104.36=c^2 \\ \sqrt[]{104.36=c} \\ c=10.21 \\ \text{ANSWER: 10.21 Inches} \end{gathered}[/tex]19.Solve the inequality. Express your answer in the form of a graph and in interval notation. (x-3) / (x+6) ≤ 0
Answers
The inequality is given as,
[tex]\frac{x-3}{x+6}\leq0[/tex]Note that the denominator can never be zero otherwise the rational function would become indeterminate. So we have to exclude the value at which the denominator,
[tex]\begin{gathered} x+6=0 \\ x=-6 \end{gathered}[/tex]So the function is not defined at x = - 6.
Consider that the division of the numbers can be non-positive, only if exactly one of the numbers is non-positive.
So we have to obtain the interval in which one of the factors is positive and the other is negative.
CASE-1: When the numerator is positive and the denominator is negative,
Graph the solution of inequalities and shade in the solution set. If there are no solutions, graph the corresponding line and do not shade in any region. Y = grater and equal to 1/2x - 3 Y < - 2/3 x + 2
Answers
start sketching the first inequality taking into account that it is all the region above the line of slope 1/2 and crosses over -3
then, do the same for the second equation over the same plane and look at the intersection
the shaded region should be the triangle at the left since the region is a solution to both inequalities.
Find the polynomial that represents the perimeter of the figure. simplify your answer.
Answers
The given diagram is a pentagon with different side measurements.
The perimeter is defined as the sum of all external boundaries of the figure.
So the perimeter (P) of the pentagon is equal to the sum of the 5 sides of the figure,
[tex]\begin{gathered} P=(3t^2-9)+(3t^2-9)+(2t^2+5)+(2t^2+5)+(t^3-t^2+8) \\ P=3t^2-9+3t^2-9+2t^2+5+2t^2+5+t^3-t^2+8 \\ P=t^3+(3t^2+3t^2+2t^2+2t^2-t^2)+(-9-9+5+5+8) \\ P=t^3+9t^2+(0) \\ P=t^3+9t^2 \end{gathered}[/tex]Thus, the perimeter of the figure is,
[tex]t^3+9t^2[/tex]Solve: 5(x + 3) = 2x -3
Answers
we need to isolate x
first solve the parenthesis
[tex]5x+15=2x-3[/tex]now solve x
[tex]\begin{gathered} 5x-2x=-3-15 \\ 3x=-18 \\ x=-\frac{18}{3} \\ x=-6 \end{gathered}[/tex]the value of x is -6
Find the diameter of the circle.The image shows a circle with radius 9 centimeters.The diameter of the circle is _______ centimeters.The solution is _______
Answers
The diameter of the circle is given as
[tex]\begin{gathered} \text{diameter}=2\times radius\text{ of the circle} \\ d=2r \end{gathered}[/tex]The given radius is
[tex]=9\operatorname{cm}[/tex]Therefore, the diameter will be
[tex]\begin{gathered} \text{diameter =2}\times9\operatorname{cm} \\ \text{diameter =18cm} \end{gathered}[/tex]Hence,
The diameter of the circle =18 centimeters
Use it in a lot and it can take forever
Answers
Part 1
Remember that
The rate of change is the same that the first derivative
so
The rate of change appears to be zero at times
t=1 weeks
t=4 weeks
t=6 weeks
Part 2
which of the following is true?
Verify each option
W'(3) > W'(7) -----> is not true
W'(3) < W'(7) -----> is trueW'(3)=W'(7) is not true
which of the following is true?
Verify each option
W'(0) < W'(5) -----> is truetherefore
the graph is
I need help graphing and I need to know the coordinates. The graph goes up to 12.
Answers
We have the next given function:
[tex]f(x)=\sqrt[3]{x}-2[/tex]To find the first point, we need to use:
[tex]\sqrt[3]{x}=0[/tex]Solve the equation for x:
[tex](\sqrt[3]{x})^3=(0)^3[/tex][tex]x=0[/tex]So, when x=0, we got the first point (0, -2), because:
[tex]y=\sqrt[3]{x}-2[/tex][tex]y=0-2[/tex]Then
[tex]y=-2[/tex]Let's find the points on right, let use x=8 and x=27
Replace on the function, when x=8
[tex]y=\sqrt[3]{x}-2[/tex][tex]y=\sqrt[3]{8}-2[/tex][tex]y=2-2[/tex][tex]y=0[/tex]So, it represents the point (8,0)
Now, when x=27
[tex]y=\sqrt[2]{27}-2[/tex][tex]y=3-2[/tex][tex]y=1[/tex]This corresponds to the point (27,1)
Now, for points on the left side:
When x=-8
[tex]y=\sqrt[3]{-8}-2[/tex][tex]y=-2-2=-4[/tex]Which represents the point (-8,-4)
When x=-27
[tex]y=\sqrt[3]{-27}-2[/tex][tex]y=-3-20-5[/tex]Which represents the point (-27, -5)
Finally, graph these four points on the cartesian plane.
Consider the fraction 1/2, if this fraction is divided by 3, will the quotient be more or less than the quotient in the first part?
Answers
What is the x-value of the solution to this system of equations? 6x + 8y = -18x = -2y - 5
Answers
Since the second equation is x in function of y, we can use the substitution method to find y and then find x
First, we substitute x from the second equation into the first equation:
[tex]\begin{gathered} 6(-2y-5)+8y=-18 \\ -12y-30+8y=-18 \end{gathered}[/tex]And solve for y:
[tex]\begin{gathered} (-12+8)y-30=-18 \\ -4y=-18+30 \\ -4y=12 \\ y=\frac{12}{-4}=-3 \end{gathered}[/tex]And now we replace y = -3 into the second equation:
[tex]\begin{gathered} x=-2y-5 \\ x=-2\cdot(-3)-5 \\ x=6-5 \\ x=1 \end{gathered}[/tex]The x-value of the solution is 1
The following data for a random sample of banks in two cities represent the ATM fees for using another bank's ATM. Compute the range and sample standard deviation for ATM feesfor each city. Which city has the most dispersion based on range? Which city has more dispersion based on the standard deviation?City A 2.50 1.50 1.25 0.00 2.00City B 1.25 1.00 1.50 1.00 1.00
Answers
Given:
City A: 2.50 1.50 1.25 0.00 2.00
City B: 1.25 1.00 1.50 1.00 1.00
To find dispersion based on range:
The difference between the maximum and minimum values in a set of data is the range.
2.5 is the maximum value of data of city A and 0.00 is the minimum value of data of city A.
Hence, the range of data of city A is,
[tex]R_A=2.50-0.00=2.50[/tex]1.5 is the maximum value of data of city B and 1.00 is the minimum value of data of city B.
Hence, the range of data of city B is,
[tex]R_B=1.50-1.00=0.50[/tex]Since the range of city A is greater than that of city B, city A has most dispersion based on range.
To find dispersion based on the standard deviation:
The data for city A is,
2.50 1.50 1.25 0.00 2.00
The mean of city A is,
[tex]\begin{gathered} \mu_A=\frac{2.5+1.5+1.25+0+2}{5} \\ =1.45 \end{gathered}[/tex]Let each individual value is represented by xi. Then, the squared difference of each individual value of city A is,
[tex](x_i-\mu_A)^2[/tex]Now, find the square of difference of each individual value of city A is,
[tex]\begin{gathered} _{}(2.5-1.45)^2=(1.05)^2=1.1025 \\ (1.5-1.45)^2=(0.05)^2=0.0025 \\ (1.25-1.45)=(-0.2)^2=0.04 \\ (0.00-1.45)^2=(-1.45)^2=2.1025 \\ (2.00-1.45)^2=(0.55)^2=0.3025 \end{gathered}[/tex]The number of values in the data set is n=5.
Let each individual value is represented by xi, then the sample standard deviation is,
[tex]S_A=\frac{1}{n-1}\sum ^n_{i\mathop=1}(x_i-\mu_A)^2[/tex]Hence, the sample standard deviation of city A can be calculated as,
[tex]\begin{gathered} S_A=\sqrt{\frac{1}{5-1}(1.1025+0.0025_{}+0.04+2.1025+0.3025)}_{} \\ =\sqrt[]{\frac{3.55}{4}} \\ =0.9420 \end{gathered}[/tex]Therefore, the sample standard deviation of city A is 0.9420.
The data for city B is:
1.25, 1.00, 1.50, 1.00, 1.00
The mean of city B is,
[tex]\begin{gathered} \mu_B=\frac{1.25+1.00+1.50+1.00+1.00}{5} \\ =1.15 \end{gathered}[/tex]Let each individual value is represented by xi. Then, the squared difference of each individual value of city B is,
[tex]\begin{gathered} (x_i-\mu)^2 \\ (1.25-1.15)^2=0.1^2=0.01 \\ (1.00-1.15)^2=(-0.15)^2=0.0225 \\ (1.5-1.15)^2=(0.35)^2=0.1225 \\ (1.00-1.15)^2=(-0.15)^2=0.0225 \\ (1.00-1.15)^2=(-0.15)^2=0.0225 \end{gathered}[/tex]The number of values in the data set is n=5.
Let each individual value is represented by xi, then the sample standard deviation of city B is,
[tex]S_B=\frac{1}{n-1}\sum ^n_{i\mathop=1}(x_i-\mu_B_{})^2[/tex]Hence, the sample standard deviation of city B can be calculated as,
[tex]\begin{gathered} S_B=\sqrt[]{\frac{1}{5-1}(0.01+0.0225_{}+0.1225+0.0225+0.0225)}_{} \\ =\sqrt[]{\frac{0.2}{4}} \\ =0.2236 \end{gathered}[/tex]Therefore, the sample standard deviation of city B is 0.2236.
Since the standard deviation of city A is greater than that of city B, city A has more dispersion based on the standard deviation.
The points H(-8,-1),I (-6,-9), J (-2,-8) and K (-4,0) form a quadrilateral. Find the desired slopes and lengths, then fill in the words that best identifies the type of quadrilateral
Answers
Answer:
[tex]\text{Quadrilateral HJLK is a }Rec\tan gle[/tex]Explanation:
Here, we want to find the slopes and lengths of the sides of a quadrilateral
To find the slopes, we use the equation:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]To find the length, we use the equation:
[tex]L\text{ = }\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]We take the sides one after the other
a) HI
We have the slope as:
[tex]m\text{ = }\frac{-9+1}{-6+8}\text{ = }\frac{-8}{2}\text{ = -4}[/tex]We have the length as:
[tex]\begin{gathered} \sqrt[]{(-6+8)^2(-9+1)^2} \\ =\text{ }\sqrt[]{4+64} \\ =\text{ }\sqrt[]{68} \end{gathered}[/tex]b) IJ
We have the slope as:
[tex]m\text{ = }\frac{-8+9}{-2+6}\text{ = }\frac{1}{4}[/tex]We have the length as:
[tex]\begin{gathered} IJ\text{ = }\sqrt[]{(-6+2)^2+(-8+9)^2} \\ IJ\text{ = }\sqrt[]{17} \end{gathered}[/tex]c) JK
Slope:
[tex]m\text{ = }\frac{-8+0}{-2+4}\text{ = -4}[/tex]Length:
[tex]\begin{gathered} JK\text{ = }\sqrt[]{2^2+(-8)^2} \\ JK\text{ = }\sqrt[]{68} \end{gathered}[/tex]D) KH
Slope:
[tex]m\text{ = }\frac{0+1}{-4+8}\text{ = }\frac{1}{4}[/tex]Length:
[tex]\begin{gathered} KH\text{ = }\sqrt[]{(-4+8)^2+(0+1)^2} \\ KH\text{ = }\sqrt[]{17} \end{gathered}[/tex]From the answers obtained, the side lengths KH and IJ are the same, while the side lengths JK and KI are the same
Also, looking at the slopes, when the product of the slopes of two lines equal -1, the two lines are perpendicular
Since:
[tex]\frac{1}{4}\times\text{ (-4) = -1}[/tex]We can conclude that a set of two sides(KH, JK and HI, IJ) are perpendicular
Thus, we have it that the quadrilateral is a rectangle
log2 z + 2 log2 x + 4 log, y + 12 logg x - 2 log2 y
Answers
apply the property of the potency for those that have coefficients
[tex]b\cdot\log x=\log (x^b)[/tex]apply the product property and the quotient product to leave it as a single log
[tex]\begin{gathered} \log (a)+\log (b)=\log (a\cdot b) \\ \log (a)-\log (b)=\log (\frac{a}{b}) \end{gathered}[/tex]simplify the expression using this properties
[tex]\begin{gathered} \log _2(z)+2\log _2(x)+4\log (y)+12\log (x)-2\log _2(y) \\ \log _2(z)+\log _2(x^2)+\log (y^4)+\log (x^{12})-\log _2(y^2) \\ \log _2(x^2z)+\log (x^{12}y^4)-\log _2(y^2)^{} \\ \log _2(\frac{x^2z}{y^2})+\log (x^{12}y^4) \end{gathered}[/tex]