Consider the given figure, in the triangle ADE and ABC,
Angle A is common in both the triangles.
Already given that angle ADE is equal to angle ABC.
Consider the property that the sum of all three angles of a triangle is 180 degree,
[tex]\angle A+\angle ADE+\angle AED=\angle A+\angle ABC+\angle ACB\Rightarrow\angle AED=\angle ACB[/tex]Therefore, by the AAA criteria, the triangles ADE and ABC are similar.
Then the sides of the triangle are proportional,
[tex]\frac{AD}{DB}=\frac{AE}{EC}[/tex]This can also be written as,
[tex]AD\colon DB=AE\colon EC[/tex]This is the required answer.
Two markers A and B on the same side of a canyon rim are 56 feet apart. A third marker C, located across the rim. is positioned so that BAC = 69º and ABC = 51° Complete parts (a) and (b) below (a) Find the distance between C and A.
To answer this question, it will be helpful to have a drawing of the situation to find the asked distance:
With this information, it will be easier to have all the information to solve for the distance CA.
Therefore, to find the distance CA, we can apply the Law of Sines, in which we have to find the angle C. We know that the sum of the interior angles of a triangle is equal to 180. Then, we have:
[tex]mNow, we can apply the Law of Sines to find the distance CA:[tex]\frac{AC}{\sin(51)}=\frac{56}{\sin(60)}\Rightarrow AC=\frac{56\cdot\sin (51)}{\sin (60)}[/tex]Then, we have:
[tex]AC=50.2527681652ft[/tex]Then, to round to one decimal place, we have that AC is approximately 50.3 ft.
To find the distance between the two rims, we have:
Now, we can also apply the Law of Sines to find the distance CD (the distance between the two rims):
[tex]\frac{CD}{\sin(69)}=\frac{CA}{\sin(90)}\Rightarrow CD=CA\cdot\sin (69),\sin (90)=1[/tex]Then, we have:
[tex]CD=50.2527681652\cdot\sin (69)\Rightarrow CD=46.9150007363ft[/tex]Therefore, the distance between the two canyon rims (round to one decimal place) is 46.9 ft.
If we take 50.3 ft (for CA), instead, we have 47 ft.
classify the systems of equations as consistent dependent, consistent independent,or inconsistent
Recall that:
1) A system of 2 equations is inconsistent if both equations represent different parallel lines.
2) A system of 2 equations is consistent dependent if the equations are equivalent.
3) A system of 2 equations is consistent independent if the slopes of both equations are different.
A) Multiplying the second equation by 2 we get:
[tex]\begin{gathered} \frac{1}{2}y\times2=(x-2)\times2, \\ y=2x-4. \end{gathered}[/tex]Notice that the above equation is the same as the first equation, therefore the equations of the first system of equations are equivalent, then the system is consistent dependent.
B) Notice that the slope of both equations is 4, also, notice that the y-intercept of the first equation is (0,2), and the y-intercept of the second equation is (0,-3), therefore the equations of the system of equations represent different parallel lines, then the system is inconsistent.
C) Notice that the slope of the first equation is 5 and the slope of the second one is 6, therefore the system of equations is consistent independent.
Answer:
A) Consistent dependent.
B) Inconsistent.
C) Consistent independent.
936.1 ÷ 2.3how do i calculate this without a calculator
Using long division:
Move the decimal point in the divisor and the dividend 1 unit
If AC = 66, find the value of x. Round your answer to the nearest tenth if necessary.AB = 8x - 25BC = 9x - 17
We need to represent the segments in a like, like in the following image:
From the image we can see that the sum of the segments AB and BC must be equal to the whole
Order: ABC 175 mg po. Stock ABC 350 mg po scored tablets. How many tablets would patient take per dose?
The number of tablets that the individual would take per dose would be = 0.5 tablet.
What is a drug?A drug is a substance that is usually prescribed by a physician which when taken has the ability to alter the physiological condition of an individual.
The order or prescribed dosage of the drug ABC = 175mg / dose
The vehicle measurement of the drug = 350mg/tab
If 1 tablet = 350 mg
X tablet = 175 mg
Make X tablet the subject of formula;
X tablet n= 175/350
X tablet = 0.5 tablet or 1/2 tablet.
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For what values of x is the expression below defined?A.-5 x < 1B.5 > x -1C.5 > x > 1D.5 x 1
Given:
There are given that the expression:
[tex]\frac{\sqrt{x+5}}{\sqrt{1-x}}[/tex]Explanation;
First, let's notice that we need positives to numbers inside both roots.
So,
The root of a negative number is a math error.
Then,
With that information, let us analyze the options.
From option A:
If we add 5 to this inequality, we have:
[tex]\begin{gathered} -5+5\leq x+5<1+5 \\ 0\leq x+5<6 \end{gathered}[/tex]That means the number in the first root is positive.
Now, we want 1-x to be positive:
[tex]\begin{gathered} -5\leq x<1 \\ 5\ge-x>-1 \\ 1+5\ge1-x>1-1 \\ 6\ge1-x>0 \end{gathered}[/tex]So, it is positive:
Final answer;
Hence, the correct option is A.
Write an equation in slope-intercept form that contains the points (2, 8) and (4, 9).
Given two points, the equation of the line in slope form can be obtained using this equation
[tex]\frac{y_2-y_1}{x_2-x_1}\text{ = }\frac{y_{}-y_1}{x_{}-x_1}[/tex]Now we can name the points
x1 = 2, y1 = 8
x2 = 4 , y2 =9
These coordinates can then be substituted into the equation
[tex]\frac{9-8}{4-2}\text{ =}\frac{y\text{ - 8}}{x\text{ - 2}}[/tex][tex]\begin{gathered} \frac{1}{2}\text{ = }\frac{y\text{ - 8}}{x\text{ - 2}} \\ \\ x-2\text{ = 2 (y - 8)} \\ \\ x\text{ - 2 = 2y - 16} \end{gathered}[/tex]x - 2 + 16 = 2y
2y = x - 2 +16
2y = x + 14
Divide both sides by 2
y = x/2 + 14/2
[tex]y\text{ = }\frac{x}{2}\text{ + 7}[/tex]This is the equation in slope-intercept form
where the slope = 1/2
Use the table. What percentage of the people surveyed were teachers who wanted a later start time?
The Solution.
The percentage of the people survey that were teachers that voted yes to start later is
[tex]\text{ }\frac{\text{ number of teachers that voted YES}}{\text{ Total number of people surveyed}}\times100[/tex]Which is
[tex]\frac{20}{75}\times100=0.266667\times100=26.6667\approx26.67\text{ \%}[/tex]b. The percentage of the people surveyed that were teachers is
[tex]\frac{\text{ number of teachers surveyed}}{\text{ Total number of people surveyed}}\times100[/tex]Which is
[tex]\frac{30}{75}\times100=0.4\times100=40\text{ \%}[/tex]Hence, the correct answer are:
a. 26.67% b. 40%
What is the image of (2,-3) after a 180 degree counterclockwise rotation about the origin?a. (-3, 2) b.(-2, 3) c. (-3, -2)d.(-2,3)
Answer:
b.(-2, 3)
Explanation:
A 180 roration transforms the coordinates of a point according to the following rule.
[tex](x,y)\rightarrow(-x,-y)[/tex]For our point (2, -3), applying the above rule gives.
[tex](2,-3)\rightarrow(-2,3)[/tex]Hence, the coordinates of the image are (-2, 3 ) which is choice B.
please explain briefly..limits and derivatives
The logarithmic-radical expression √[㏒ₐ f(x)] is true for 0 < f(x) ≤ 1. (Correct choice: D)
What is the domain of a logarithmic-radical function?
Logarithms are trascendent expressions whose domain is described below:
Ran (logₐ f(x)) = (0, + ∞)
Since 0 < a < 1, then we find the following feature: logₐ f(x) > 0 for 0 < f(x) ≤ 1.
In addition, the domain of radical functions is described below:
Dom (√f(x)) = f(x) ≥ 0
Therefore, the logarithmic-radical expression defined in the statement is true for 0 < f(x) ≤ 1.
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trig The last sub-problem of this section stumped me pls help
For this problem, we are given a triangle and we need to determine its height.
The distance of the UFO from point A is equal to the side c of the triangle, this side forms a right triangle with the height, where the height is the opposite cathetus from angle alpha and side c is the hypothenuse. We can use the sine relationship to determine the height, as shown below:
[tex]\begin{gathered} \sin(87.4)=\frac{h}{425.58}\\ \\ h=425.58\cdot\sin(87.4)\\ \\ h=425.58\cdot0.9989706=425.14 \end{gathered}[/tex]The height is approximately 425.14 km.
Find a measurement of the complement for the angle 20
Given:
There are given that the angle is 20 degrees.
Explanation:
According to the concept:
The complementary angle is:
[tex]90^{\circ}-\theta[/tex]Then,
Put the value of an angle;
So,
[tex]\begin{gathered} 90^{\circ}-\theta=90^{\circ}-20 \\ =70^{\circ} \end{gathered}[/tex]Final answer:
Hence, the measure of the complement is 70 degrees.
There is 1/5 of a foot of ribbon left onthe spool. If Brittany cuts it into 3equal pieces, how long (in feet) willeach piece be?
We know that
• There is 1/5 of a foot of ribbon.
If Brittany cuts it into 3 equal pieces, we have to divide to find the length of each piece.
[tex]\frac{\frac{1}{5}}{3}=\frac{1}{15}[/tex]Therefore, each piece is 1/15 of a foot long.With the exception of column one, all amounts are in dollars. Calculate the annual interest rate on this loan. Give your answer to the nearest hundredth percent. Do not include the % sign in your response.
Given:
Amortization table is given
Let r be the annual rate of interest.
[tex]\frac{r}{12}\text{ be the monthly rate of interest.}[/tex]Second payment:
P= $259873.20 ; interest = $539.24
[tex]\text{Interest for the 2nd payment = }P(\frac{r}{12}\times\frac{1}{100})[/tex][tex]539.24=259873.20(\frac{r}{1200})[/tex][tex]\frac{539.24}{259873.20}\times1200=r[/tex][tex]r=\frac{647088}{259873.20}[/tex][tex]r=2.49[/tex]Therefore, the annula rate of interest is 2.49%
Sanjay attempts a 50-yard field goal in a football game. For his attempt to be a success, the football needs to pass through the uprights and over the crossbar that is 10 feet above the ground.Sanjay kicks the ball from the ground with an initial velocity of 64 feet per second, at an angle of 34° with the horizontal.Is Sanjay's attempt successful? If not, how many feet too low is the ball?
Let us draw a sketch to understand the situation
We will use some rules here
[tex]\begin{gathered} v_x=vcos\theta=64cos(34) \\ d_x=v_xt=64cos(34)t \end{gathered}[/tex]Since the horizontal distance is 50 yards
Since 1 yard = 3 feet, then
[tex]d_x=50\times3=150feet[/tex]We will use it to find the time t
[tex]\begin{gathered} d_x=150 \\ 64cos(34)t=150 \\ t=\frac{150}{64cos(34)}\text{ s} \end{gathered}[/tex]Now, we will find the vertical distance (h) by using this rule
[tex]\begin{gathered} v_y=vsin\theta=64sin(34) \\ d_y=h=v_yt-\frac{1}{2}at^2=64sin(34)t-\frac{1}{2}(32)t^2 \end{gathered}[/tex]Note that: a is the acceleration of gravity which is 32 ft/s^2
We will substitute t by its value
[tex]h=64sin(34)(\frac{150}{64cos(34)})-16(\frac{150}{64cos(34)})[/tex]We can simplify it by using sin34/cos34 = tan34, and 1/cos34 = sec34
But I will put it on the calculator to find the final answer
[tex]h=55.94\text{ ft}[/tex]Since the height of the crossbar is 10 feet, then
Sanjay's attempt successful
Leila bought a sofa on sale for $268. This price was 33% less than the original price.What was the original price?
Let P be the original price.
Since $268 is 33% less than the original price, then $268 is equal to 67% of the original price:
[tex]268=\frac{67}{100}\times P[/tex]Then:
[tex]\begin{gathered} P=\frac{100}{67}\times268 \\ =400 \end{gathered}[/tex]Therefore, the original price was $400.
Answer: $356.44
Step-by-step Explanation: To find the original price of the sofa you need to multiply 33% by $268, but you need to turn the percent into a decimal, to do so you need to divide 33 by 100 & that is 0.33. So 0.33 x $268 is 88.44. After, you add both $268 and $88.44 to get the original price & that is $356.44.
Translate to a system of equations. Do not solve.Two angles are supplementary. One angle is 4 less than three times the other . Find the measures of the angles l.
Two angles are supplementary
That means they add to 180
x+y = 180
One angle is 4 less than three times the other
We know that is means equals and less than comes after
x = 3y-4
Which of the following equations does the graph below represent?
A. 2x + 2y = 8
B. -2x - 2y = 8
C. -2x + y = 8
D. -2x + 2y = 8
Answer: D
Step-by-step explanation:
The answer is D, as seen on the graph, the Y-Intercept is at Y = 4, and the gradient is 1, so according to the equation y = mx + c,
"m" must equal 1, and "c" must equal 4, so the equation needs to be:
y = x + 4.
In Option D, the equation can be rearranged to 2y = 2x + 8, dividing both the LHS and RHS by 2, we get y = x + 4.
This type of question can be tough at first, however it's just a matter of practice, keep practicing, keep working hard, and you'll be an expert in no time!
Solve for the remaining angles and side of the two triangles that can be created. Round to the nearest hundredth:B = 30 .b = 6,a = 7AnswerHow to enter your answer (opens in new window) 2 PointsTriangle 1: (where angle A is acute):Triangle 2: (where angle A is obtuse):AA:C =C:C:
ANSWER:
Triangle 1:
A = 35.69°
C = 114.31°
c = 10.94
Triangle 2:
A = 144.31°
C = 5.69°
c = 1.19
STEP-BY-STEP EXPLANATION:
Given:
B = 30°, b = 6, a = 7
We calculate the angle A by means of the law of sines:
[tex]\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B} \\ \\ \text{ We replacing} \\ \\ \frac{7}{\sin A}=\frac{6}{\sin30} \\ \\ \sin A=\frac{7}{6}\cdot\sin30 \\ \\ \sin A=\frac{7}{12} \\ \\ A=\sin^{-1}\left(\frac{7}{12}\right)\: \\ \\ A_{acute}=35.69\degree \\ \\ A_{obtuse}=144.31\degree \end{gathered}[/tex]We calculate the value of angle C, knowing that the sum of all internal angles is equal to 180°
[tex]\begin{gathered} \text{ Acute} \\ \\ 180=35.69+30+C \\ \\ C=180-30-35.69=114.31\degree \\ \\ \text{ Obtuse} \\ \\ 180=144.31+30+C \\ \\ C=180-30-144.31=5.69\degree \end{gathered}[/tex]Side c is also calculated with the law of sines, like this:
[tex]\begin{gathered} \text{ Acute} \\ \\ \frac{b}{\sin B}=\frac{c}{\sin C} \\ \\ \frac{6}{\sin(30)}=\frac{c}{\sin114.31} \\ \\ c=\frac{6}{\sin(30)}\cdot\sin114.31 \\ \\ c=\:10.94 \\ \\ \text{ Obtuse} \\ \\ \frac{7}{\sin(A)}=\frac{c}{\sin(C)} \\ \\ c=\frac{6}{\sin(30)}\sin(5.69) \\ \\ c=1.19 \end{gathered}[/tex]Therefore;
Triangle 1:
A = 35.69°
C = 114.31°
c = 10.94
Triangle 2:
A = 144.31°
C = 5.69°
c = 1.19
Solve the given expression for x = -18:5x/3 - 2
ANSWER
[tex]-32[/tex]EXPLANATION
We want to solve the given expression for x = -18:
[tex]\frac{5x}{3}-2[/tex]To do this, substitute the given value of x into the expression and simplify. That is:
[tex]\begin{gathered} \frac{5(-18)}{3}-2 \\ \frac{-90}{3}-2 \\ -30-2 \\ \Rightarrow-32 \end{gathered}[/tex]That is the answer.
How do I solve these?If f(x)=3xsquared + 9x-4 then evaluate the following:f(1)=3x^2+9x-4f(x+h)=3x^2+9x-4
a) We need to evaluate when x = 1
f(1): this means we will replace x with 1 in the given function
[tex]\begin{gathered} f\mleft(x\mright)=3x^2+9x-4 \\ f\mleft(1\mright)=3(1)^2+9(1)-4 \\ f(1)\text{ = 3(1) + 9 - 4 = 3 + 9 - 4} \\ f(1)\text{ = 8} \end{gathered}[/tex]b) We need to evaluate the function when x = x + h
[tex]\begin{gathered} f\mleft(x\mright)=3x^2+9x-4 \\ f(x\text{ + h): we will replace x with x + h in the given function} \\ f(x+h)=3(x+h)^2\text{ + 9(x + h) - 4} \end{gathered}[/tex]Expanding:
[tex]\begin{gathered} f(x\text{ + h) }=3(x^2+2xh+h^2)\text{ + 9(x + h) - 4} \\ f(x\text{ + h) }=3x^2+6xh+3h^2\text{ + 9x + 9h - 4} \\ \text{Since there are no like terms we can simplify, we can leave it in expanded form:} \\ f(x\text{ + h) }=3x^2+6xh+3h^2\text{ + 9x + 9h - 4} \\ \\ or\text{ the non expanded form:} \\ f(x+h)=3(x+h)^2\text{ + 9(x + h) - 4} \end{gathered}[/tex]Write anequivalent expression by distributing thesign outside the parentheses:-(2h + 9.6k) +1
The given expression is
-(2h + 9.6k) +1
Due to the - sign outside the parentheses, every + sign inside the parentheses would be changed to a - sign. Thus, the expression becomes
- 2h - 9.6k + 1
The mean of the following data values is 32. 19, 23, 35, 41, 42A. True B. False
Remember that mean of a set is another name for the average of that set. To find the mean of a data set, add all the values together and divide by the number of values in the set.
Thus, if we have the following set of values: 19, 23, 35, 41, 42, the mean would be:
[tex]\frac{19\text{ + 23 + 35 + 41 + 42}}{5}=\frac{160}{5}=32[/tex]The correct answer:
Answer:TRUE
limit using L'Hopital's rule . I just want to make sure if my answer is correct or not?
In order to use L'Hopital's rule, it is necessary to rewrite the limit as the quotient of two functions. Notice that:
[tex]\begin{gathered} 6x^{\sin (4x)}=e^{\ln (6x^{\sin (ex)})^{}} \\ =e^{\sin (4x)\cdot\ln (6x)} \end{gathered}[/tex]Since the exponential function is a continuous function, then:
[tex]\lim _{\text{x}\rightarrow0}e^{\sin (4x)\cdot\ln (6x)}=e^{\lim _{x\rightarrow0}\sin (4x)\cdot\ln (6x)}[/tex]Find the following limit using L'Hopital's rule:
[tex]\lim _{x\rightarrow0}\sin (4x)\cdot\ln (6x)[/tex]Write the function as a fraction:
[tex]\lim _{x\rightarrow0}\frac{\ln (6x)}{(\frac{1}{\sin (4x)})}[/tex]Use L'Hopital's rule to rewrite the limit as the limit of the quotient of the derivatives:
[tex]\begin{gathered} \lim _{x\rightarrow0}\frac{(\frac{1}{x})}{(-\frac{4\cos(4x)}{\sin^2(4x)})}=\lim _{x\rightarrow0}-\frac{\sin ^2(4x)}{4x\cdot\cos (4x)} \\ =\lim _{x\rightarrow0}\sin (4x)\cdot\frac{\sin(4x)}{4x}\cdot\frac{-1}{\cos (4x)} \\ =\lim _{x\rightarrow0}\sin (4x)\cdot\lim _{x\rightarrow0}\frac{\sin(4x)}{4x}\cdot\lim _{x\rightarrow0}\frac{-1}{\cos (4x)} \\ =0\cdot1\cdot-1 \\ =0 \end{gathered}[/tex]Therefore:
[tex]\lim _{x\rightarrow0}6x^{\sin (4x)}=e^0=1[/tex]Crystal earns $4.75 per hour mowing lawns. A. write a rule to describe how the amount of money M earned is a function of the number of hours H that mowing lawns. B. l how much does crystal earn if she works 1 hour and 15 minutes?
Given:
Crystal earns $4.75 per hour mowing lawns.
Let the money earned = M
And the number of hours = H
So, the relation between M and H will be :
[tex]M=4.75\cdot H[/tex]B. how much does crystal earn if she works 1 hour and 15 minutes?
Time = 1 hours and 15 minutes
AS 1 hour = 60 minutes
So,
[tex]H=1+\frac{15}{60}=1+\frac{1}{4}=1+0.25=1.25[/tex]Substitute with H to find M
So,
[tex]M=4.75\cdot1.25=5.9375[/tex]So, she will earn $5.9375
solve the system by subsitution method
Substitute Y = 3X - 6
in second equation
-15X + 5•(3X - 6) = -30
Now solve for X, cancel parenthesis
use a(b+c) = ab + ac
-15X + 15 X - 30 = -30
. -30 = -30
Then we see that, have infinite solutions
In consecuence, ANSWER IS
OPTION D) (x , 3x - 6 )
SCC Library667737985Based on the graph of this normal distribution,a. The mean isb. The median isThe mode isd. The standard deviation isCheck Answer
The Solution.
From the graph,
a. The mean = 73
b. The median = 73
c. The mode = 73
d. The standard deviation (S.D) is;
[tex]S.D=73-67=6[/tex]set up an equation for your exterior angle, then use multi-step equation steps to solve for y.A. 15B. 17.4C. 5D. 10
In any triangle, the sum of the interior angles of two vertices is equal to the exterior angle of the other vertex.
Using this property, we can write the following equation:
[tex]\begin{gathered} \text{ABC+BAC=ACD}_{} \\ (4y+8)+(5y+3)=146 \\ 9y+11=146 \\ 9y=146-11 \\ 9y=135 \\ y=\frac{135}{9} \\ y=15 \end{gathered}[/tex]The value of y is equal to 15, therefore the correct option is A.
solve the equation for all values of x by completing the square. x²+8x=-15
since (8/2)^2=16, we will add 16 in both sides of the equation, obtaining
[tex]x^2+8x+16=1[/tex]now, we factor the left side of the equation (it's a perfect square)
[tex](x+4)^2=1[/tex]then we have two options or x+4=1 or x+4=-1
solving both of the we have that the values for x are x=-3 and x=-5Let's test out the prediction! On the coordinate plane below, plot the points from your table in Slide 4 and sketch the graph.Table from slide 4: Bounce Height after Bounce 1. 92. 8.13. 7.294. 6.561
Answer
Check Explanation
Explanation
To do this, we will let the bounce be represented on the x-axis as x and the height after bounce plotted on the y-axis as y
So, the table looks like
x | y
1 | 9
2 | 8.1
3 | 7.29
4 | 6.561
So, we plot these points on a graph and sketch a line of best fit to pass through them
Hope this Helps!!!