The product of the slopes of the perpendicular lines is -1, which means if the slope of one of them is m, then the slope of the perpendicular line is -1/m
In triangle ABC
The perpendicular bisector of the side BC is drawn from the opposite vertex A
Then to find it find the slope of BC and reciprocal it and change its sign to get its slope and find the midpoint of BC to use it in the equation of the perpendicular bisector
Since B = (7, 3) and C = (4, 6)
Let us find the slope of BC, using the rule of the slope
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Let (x1, y1) = (7, 3) and (x2, y2) = (4, 6)
[tex]\begin{gathered} m_{BC}=\frac{6-3}{4-7} \\ m_{BC}=\frac{3}{-3} \\ m_{BC}=-1 \end{gathered}[/tex]Now to find the slope of the perpendicular line to BC reciprocal it and change its sign
Since the reciprocal of 1 is 1 and the opposite of negative is positive, then
Then the slope of the perpendicular line is 1
Now, let us find the mid-point of BC
The rule of the midpoint is
[tex]M=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]Then the mid-point of BC is
[tex]\begin{gathered} M_{BC}=(\frac{7+4}{2},\frac{3+6}{2}) \\ M_{BC}=(\frac{11}{2},\frac{9}{2}) \\ M_{BC}=(5.5,4.5) \end{gathered}[/tex]Now we can form the equation of the perpendicular bisector of BC using its slope 1 and the point (5.5, 4.5)
The form of the equation using a point and a slope is
[tex]y-y1=m(x-x1)[/tex]m is the slope and (x1, y1) is a point on the line
Since m = 1 and (x1, y1) = (5.5, 4.5), then
[tex]\begin{gathered} m=1,x1=5.5,y1=4.5 \\ y-4.5=1(x-5.5) \\ y-4.5=x-5.5 \end{gathered}[/tex]Add 4.5 to both sides
[tex]\begin{gathered} y-4.5+4.5=x-5.5+4.5 \\ y=x-1 \end{gathered}[/tex]The equation of the perpendicular bisector of BC is
[tex]y=x-1[/tex]We will do the same to AB and AC
-ractions:
On a website, there is an ad for jeans every 5 minutes, an ad for sneakers
every 10 minutes, and an ad for scarves every 45 minutes.
If they all appeared together at 9:00 P.M., when is
the next time they will all appear together?
ICM to solve the problem
Answer:
Step-by-step explanation:
Write the equation of a line the goes through point
(3,-4) and is perpendicular to the line x = 1.
keeping in mind that x = 1 is just a vertical line, Check the picture below.
Identify the vertex, axis of symmetry, and if the graph has a maximum or minimum. Then write the function for the graph shown
Answer:
Step-by-step explanation:
n b
6. A profit function for a new business follows the functionP(x) = 1/3x^2 - 6x, where x represents the number of months.After how many months will the company begin to make aprofit?A. 2B. 9C. 12D. 18
ANSWER
It will take 18 months before the company starts making a profit.
STEP-BY-STEP EXPLANATION
Given information
[tex]P(x)\text{ = }\frac{1}{3}x^2\text{ - 6x}[/tex]Where x is the number of months.
Step 1: Make P(x) = 0
[tex]\begin{gathered} \text{ p(x) = }\frac{1}{3}x^2\text{ - 6}x \\ 0\text{ = }\frac{1}{3}x^2\text{ - 6}x \end{gathered}[/tex]Step 2: Find x from the above equation
[tex]\begin{gathered} 0\text{ = }\frac{1}{3}x^2\text{ - 6x} \\ \text{Add 6x to the both sides} \\ 0\text{ + 6x = }\frac{1}{3}x^2\text{ - 6x + 6x} \\ 6x\text{ = }\frac{1}{3}x^2 \\ \text{cross multiply} \\ 6x\text{ }\times3=x^2 \\ 18x=x^2 \\ \text{Divide both sides by x} \\ \frac{18\cancel{x}}{\cancel{x}}\text{ = }\frac{\cancel{x^2}}{\cancel{x}} \\ x\text{ = 18 months} \end{gathered}[/tex]Therefore, it will take 18 months before the company starts making a profit.
Solve the inequality and graph the solution set.3 ≤ 4x + 1 < 9
Okay, here we have this:
Considering the provided inequality, we are going to solve it and graph the solution set, so we obtain the following:
3 ≤ 4x + 1 < 9
3 -1≤ 4x + 1 -1< 9-1
2 ≤ 4x < 8
2/4 ≤ 4x/4 < 8/4
1/2 ≤ x < 2
In interval notation the solution set will be: [1/2, 2)
And if we plot this solution interval we get:
Where the solution set will be the purple part.
I need help I need help I need help I need help I need help i need help I need help
Answer:
5) The midrange is 19.5ºF
6) The midrange is 67.5º
Explanation:
The problem tell us how to calculate the midrange.
In (5) the minimum and maximum values are given (-6ºF and 45ºF, respectively). Using the formula:
[tex]Midrange=\frac{-6+45}{2}=\frac{39}{2}=19.5ºF[/tex]In (6), we need to find the minimum and maximum values from a list of them. We can see that the minimum is 58º and the maximum 77º
Then:
[tex]Midrange=\frac{58+77}{2}=\frac{135}{2}=67.5º[/tex]Select the quadratic equation that has no real solution.9x2–25x-30 = 09x? – 25x +30 = 09x2-30x +25= 0o 9x2-30x – 25 = 0
SOLUTION:
We are to select the quadratic equation that has no real solution.
Facts about Quadratic equations;
When considering,
[tex]b^2\text{ - 4ac}[/tex]If you get a positive number, the quadratic will have two unique solutions. If you get 0, the quadratic will have exactly one solution, a double root. If you get a negative number, the quadratic will have no real solutions, just two imaginary ones.
Looking at all the four options, I have examined all and the only one found to be negative is the second option. Let's consider it together
a = 9, b = -25 and c = 30
(b x b ) - 4 x a x c
(-25 x -25) - 4 (9) (30)
625 - 1080
- 455
-455 < 0
Since the discriminant is less than this quadratic equation is expected to have no real solution.
You can as well try the other three options one is zero and the remaining two are greater than zero.
find the value of x so that AB and DC are parallel
According to the properties of a parallelogram, the consecutive interior angles are supplementary, this is that the sum of its measures is 180.
Use the expressions given for 2 of the consecutive angles to find the value of x. Remember, the sum of these expressions must be 180.
[tex]\begin{gathered} (3x+15)+(7x+25)=180 \\ 10x+40=180 \\ 10x=140 \\ x=\frac{140}{10} \\ x=14 \end{gathered}[/tex]x has a value of 14.
If f(x) = 8x2 - 18x + 5, find when f(x) = -4
Setting the given equation equals -4 we get:
[tex]\begin{gathered} 8x^2-18x+5=-4 \\ 8x^2-18x+5+4=0 \\ 8x^2-18x+9=0 \end{gathered}[/tex]Notice that:
[tex]8x^2-18x+9=8(x^2-\frac{9}{4}x+\frac{9}{8})=8(x-\frac{3}{2})(x-\frac{3}{4})[/tex]Therefore, f(x)=-4 when x=3/2 or x=3/4.
I would like to know if I have this question correct thank you
Remember that
For a 95% confidence interval --------> the value of z=1.960
Find out the value of
[tex]Z\frac{s}{\sqrt{n}}=1.960(\frac{12}{\sqrt{36}})=3.92[/tex]therefore
[tex]\begin{gathered} 230\pm3.92 \\ 230+3.92=233.92 \\ 230-3.92=226.08 \\ therefore \\ The\text{ answer is} \\ (226.08,233.92) \end{gathered}[/tex]Determine the reasonableness of a solution to a logarithmic equation
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given equation
[tex]\log_3x=7[/tex]STEP 2: State the law of logarithm
[tex]\begin{gathered} If\text{ }\log_ab=c \\ \Rightarrow b=a^c \\ By\text{ substitution,} \\ \therefore\log_aa^c=c \end{gathered}[/tex]STEP 3: Substitute the given values in the question to get the correct answer
[tex]\begin{gathered} \log_3x=7 \\ x=3^7 \\ By\text{ substitution,} \\ \log_3(3^7)=7 \end{gathered}[/tex]Hence, Answer is:
[tex]\log_3(3^7)=7[/tex]OPTION A
Find the area of the triangle.
The area of the triangle given as in the attached image to the task content is; 1 ft².
What is the area of the triangle as indicated in the attached image?It follows from the task comtent that the area of the triangle given be determined.
Since the area of a triangle is given by the formula; Area = (1/2) × base × height.
Since the base of the triangle in discuss is 3 ft and it's height (altitude) as given in the task content is; (2/3) feet.
It follows that the area is;
Area = (1/2) × 3 × (2/3).
Area = 1 ft².
Ultimately, the area of the triangle is; 1 ft².
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I need help with this question Write and expression that models the situation:Sarah has spent x dollars out of the 30 dollars she started with.
Okay, here we have this:
Considering that it says "spent", it represents an outflow of money, therefore we take it as negative, so we obtain:
Actual Situation: Initial money - money spent
Actual Situation: 30 - x
determine whether AB and AC are parallel,perpendicular,or neither.A(9,-3) , B(9,4), C(-2,10), D(-2,6)
We first determine the value of AB & CD:
AB (0, 7)
CD (0, -4)
We will calculate first if they are perpendicular:
[tex](0,7)\cdot(0,-4)=0\cdot0+(7)(-4)=-28\ne0[/tex]From this, we know AB and CD are not perpendicular.
Now, in order to know if they are parallel, we will do as follows:
[tex](0,7)x=(0,-4)[/tex]From this, we will have:
[tex](0,7x)=(0,-4)\Rightarrow7x=-4\Rightarrow x=-\frac{4}{7}[/tex]From this we have that they are multiple of each other, therefore they are parallel.
Plot the vertex of f(x) = (x − 2)2 + 2.
Take into account that the general function of a parabola in vertex form is given by:
[tex]f(x)=a(x-h)^2+k[/tex]where (h,k) is the vertex of the parabola.
By comparing the previous general function with the given function:
[tex]f(x)=(x-2)^2+2[/tex]you can notice that:
h = 2
k = 2
Hence, you can conclude that the vertex of the given function is (2,2)
9) solve using substitution method and check your answer:4x - 3y + 2z = 16- 4y - Z = 7= 146x - y
Given the system of equations, solve the third equation for y, as shown below
[tex]\begin{gathered} 6x-y=14 \\ \Rightarrow y=6x-14 \end{gathered}[/tex]And, solve for z in the second equation,
[tex]\begin{gathered} -4y-z=7 \\ \Rightarrow z=-4y-7 \\ \Rightarrow z=-4(6x-14)-7=-24x+49 \end{gathered}[/tex]Thus, substitute the values of y and z in terms of x into the first equation, as shown below
[tex]\begin{gathered} \Rightarrow4x-3y+2z=4x-3(6x-14)+2(-24x+49)=4x-18x+42-48x+98 \\ \Rightarrow-62x+140=16 \\ \Rightarrow-62x=-124 \\ \Rightarrow x=2 \end{gathered}[/tex]Then, solving for y and z given x=2,
[tex]\begin{gathered} x=2 \\ \Rightarrow y=6*2-14=-2 \\ and \\ z=-24*2+49=-48+49=1 \end{gathered}[/tex]Therefore, the solution to the system of equations is x=2, y=-2, z=1To verify the solutions, substitute the values we found into the three equations of the system, as shown below
[tex]\begin{gathered} x=2,y=-2,z=1 \\ \Rightarrow4x-3y+2z=4*2-3*(-2)+2*1=8+6+2=16\rightarrow correct \\ \Rightarrow-4y-z=-4*(-2)-1(1)=8-1=7\rightarrow correct \\ \Rightarrow6x-y=6*2-1(-2)=12+2=14\rightarrow correct \end{gathered}[/tex]Which of the following is equal to the rational expression below when x+112x² – 121x +11A. +11B.X+ 11c. -11XD. X-11
SOLUTION
From the question we have
[tex]\frac{x^2-121}{x+11}[/tex]from difference of two squares, we have
[tex]\begin{gathered} \frac{(x-11)(x+11)}{x+11} \\ x+11\text{ above cancels the one below, we have } \\ x-11 \end{gathered}[/tex]Hence the answer is option D
How many solutions does the equation 5(m + 3) = 6-7m have? Explain how you found your answer.
Expand the left hand side using distributive property:
[tex]\begin{gathered} 5\cdot m+5\cdot3=6-7m \\ 5m+15=6-7m \\ \text{Add 7m to both sides:} \\ 5m+15+7m=6-7m+7m \\ 12m+15=6 \\ \text{subtract 15 from both sides:} \\ 12m+15-15=6-15 \\ 12m=-9 \\ \text{divide both sides by 12:} \\ \frac{12}{12}m=-\frac{9}{12} \\ m=-\frac{3}{4} \end{gathered}[/tex]In a factory, the profit, P, varies directly with the inventory, I. If the factory has a profit of $60,000 when their inventory is 1,500 units, find the profit for an inventory of 50 units.
The factory has a profit of $60 000when their inventory is 1, 500 units
Let x be the profit for an inventory of 50 units
$60 000 = 1,500 units
X = 50 units
cross-multiply
1500X = $60 000 x 50
1500X =3,000,000
Divide both-side of the equation by 1500
1500X/1500 = 3,000,000/1500
x= $2000
The factory has a profit of $2000 for an inventory of 50 units
lineal or no?1) 2x+y=52) y= x + 6 --- 2thanks
1) 2x+y=5 ...... It is a linear equation
2) y= x + 6 ....... It is a linear equation
Because they are of first degree and they contain x and y (equations of a line)
Please helpwhat does A∩B=∅ mean. Thus, please help with:Suppose Pr(A)=0.3, Pr(B)=0.4 and A∩B=∅. Find:a- Pr(A∩B)b- Pr(A∪B)
Given: A and B are two sets such that-
[tex]\begin{gathered} A\cap B=\phi \\ Pr(A)=0.3 \\ Pr(B)=0.4 \end{gathered}[/tex]Required: To determine-
[tex]\begin{gathered} Pr(A\cap B) \\ Pr(A\cup B) \end{gathered}[/tex]Explanation: Since A and B have no common elements, the events are independent events or disjoints or mutually exclusive.
For independent events, we have-
[tex]Pr(A\cap B)=Pr(A).Pr(B)[/tex]Substituting the values into the formula-
[tex]\begin{gathered} Pr(A\cap B)=0.3\times0.4 \\ =0.12 \end{gathered}[/tex]Recall that-
[tex]Pr(A\cup B)=Pr(A)+Pr(B)-Pr(A\cap B)[/tex]Substituting the values into the formula and further solving as-
[tex]\begin{gathered} Pr(A\cup B)=0.3+0.4-0.12 \\ =0.7-0.12 \\ =0.58 \end{gathered}[/tex]Final Answer: a)
[tex]Pr(A\cap B)=0.12[/tex]b)
[tex]Pr(A\cup B)=0.58[/tex]Hi, can you help me answer this question please, thank you!
Consider that you have a population greater than 30, then, you can use the normal distribution to determine the margin of error.
Use the following formula:
[tex]\bar{x}\pm Z_{\frac{\alpha}{2}}\frac{s}{\sqrt[]{n}}[/tex]where:
x: mean = 33
s: standard deviation = 2
n = 31
Z: z-value for 98%
The value of Z can be found on a table for the normal distribution. For a margin of error at 98%, you get for Z:
Z = 2.326
Replace the previous values of the parameters into the formula for the margin of error (confidence interval):
[tex]\begin{gathered} 33\pm(2.326)\frac{2}{\sqrt[]{31}}= \\ 33\pm0.83 \end{gathered}[/tex]Then, the margin of error is:
(33.00 - 0.83 , 33.00 + 0.83) = (32.17 , 33.83)
The dose of a drug is critical. Too small a dose may be treat a patient effectively, A nurse must give a patient 40mg of a drug for each kilogram of the patient’s mass. If the patient weighs 165lbs how many milligrams of the drug should be given?
EXPLANATION
We need to multiply the number of needed milligrams by the weight of the patient, but first turning the weight in lbs into kilograms,
[tex]?Kilograms=165lbs*\frac{0.453}{1lb}=74.745Kg[/tex]Now, multiplying the obtained weight by the number of miligrams, give us the dose:
[tex]Dose=40\frac{mg}{Kg}*74.745Kg=2989.8mg[/tex]In conclusion, the nurse should give 2989 mg of the drug.
at Frank's auto plaza there are currently 11 new cars, 8 used cars, 12 new trucks and 10 used trucks. frank is going to choose one of these vehicles at random to be the deal of the month. what is the probability that the vehicle that frank chooses is used or is a car?
11 new cars
8 used cars
12 new trucks
10 used trucks
Total vehicles
11+8+12+10 = 41
It is the denominator of the fraction.
The subset "used" + "cars" has 11 (new cars) + 8 (used cars) + 10 (used trucks) = 29 elements.
It is the numerator of the fraction.
P(U or C) = 29/41
7. You are single and claim 1 allowance. You presently earn $319 per week.
Starting next week you will receive a 5 percent increase in pay and will earn
$335.00. How much more will you have withheld from your weekly pay for federal income tax?
I will withdraw 3 % of the of my weekly pay for federal income tax.
How do you calculate weekly pay for federal income tax?An income tax is a charge levied against people or organizations in relation to the income or profits they make. In most cases, income tax is calculated as the sum of the tax rate and the amount of taxable income. The type of taxpayer and the type of income are two factors that can affect the tax rate. Individuals (or family units) and corporations are subject to income taxes. The basis for calculating individual income tax is the income received. Since the burden is presumably on the individuals who pay it, it is typically categorized as a direct tax. Income taxes are assessed against both businesses and people based on their profits. Taxable income can be earned from a variety of sources, including earnings, salaries, dividends, interest, royalties, and rent.
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hii so i got this question wrong a while ago and im reviewing it id like some help finding out how to solve it
Answer:
Options 1, 3, and 4.
Explanation:
Given the expression:
[tex]3x\mleft(x-12x\mright)+3x^2-2\mleft(x-2\mright)^2[/tex]Step 1: The term -2(x-2)² is simplified by first squaring the expression x-2.
[tex]\begin{gathered} 3x(x-12x)+3x^2-2(x-2)^2 \\ =3x(x-12x)+3x^2-2(x-2)(x-2) \\ =3x(x-12x)+3x^2-2(x^2-2x-2x+4) \\ =3x(x-12x)+3x^2-2(x^2-4x+4) \end{gathered}[/tex]Step 2: The parentheses are eliminated through multiplication.
[tex]=3x^2-36x^2+3x^2-2x^2+8x-8[/tex]Step 3: After multiplying, the like terms are combined by adding and subtracting.
[tex]\begin{gathered} =3x^2-36x^2+3x^2-2x^2+8x-8 \\ =-32x^2+8x-8 \end{gathered}[/tex]The three options that are correct are Options 1, 3, and 4.
The vertex of a quadratic function is (2, -1) and its y-intercept is 7. Find the function,
Given:-
[tex]\text{vertex}=(2,-1),y-intercept=7[/tex]To find:-
The function.
So the formula is,
[tex]y=a\mleft(x-h\mright)^{2}+k[/tex]So substituting we get,
[tex]y=7(x-2)^2-1[/tex]So the value. we get,
[tex]\begin{gathered} y=7(x-2)^2-1 \\ y=7(x^2-4x+4)-1 \end{gathered}[/tex]Since the value of x is,
[tex]\begin{gathered} y=7x^2-28x+28-1 \\ y=7x^2-28x+27 \end{gathered}[/tex]So the value,
[tex]y=7x^2-28x+27[/tex]Consider the line y=7x-1Find the equation of the line that is perpendicular to this line and passes through the point −2, 3.Find the equation of the line that is parallel to this line and passes through the point −2, 3.Note that the ALEKS graphing calculator may be helpful in checking your answer.Equation of per pendicular line:Equation of parallel line:
Algebra / Graphs and Functions / Equations of Parallel and Perpendicular Lines
We have the line:
[tex]y=7x-1.[/tex]We must find the equation:
0. of the perpendicular line,
,1. and the parallel line,
to the given line that passes through the point (-2, 3).
1) Perpendicular line
The equation of the perpendicular line has the form:
[tex]y=m_p\cdot(x-x_0)+y_0.[/tex]Where mₚ is the slope, and (x₀, y₀) = (-2, 3).
From the equation of the given line, we see that its slope is m = 7. The slope of the perpendicular line mₚ is given by the equation:
[tex]\begin{gathered} m\cdot m_p=-1, \\ 7\cdot m_p=-1, \\ m_p=-\frac{1}{7}. \end{gathered}[/tex]Replacing mₚ = -1/7 and (x₀, y₀) = (-2, 3) in the equation of the perpendicular line, we get:
[tex]y=-\frac{1}{7}\cdot(x-(-2))+3=-\frac{1}{7}\cdot(x+2)+3=-\frac{1}{7}\cdot x-\frac{2}{7}+3=-\frac{1}{7}\cdot x+\frac{19}{7}.[/tex]2) Parallel line
The equation of the perpendicular line has the form:
[tex]y=m_p\cdot(x-x_0)+y_0.[/tex]Where mₚ is the slope, and (x₀, y₀) = (-2, 3).
From the equation of the given line, we see that its slope is m = 7. The parallel line has the same slope as the given line, so we have:
[tex]\begin{gathered} m_p=m, \\ m_p=7. \end{gathered}[/tex]Replacing mₚ = 7 and (x₀, y₀) = (-2, 3) in the equation of the parallel line, we get:
[tex]y=7\cdot(x-(-2))+3=7\cdot(x+2)+3=7x+14+3=7x+17.[/tex]3) Graph
Plotting the equations obtained, we get the following graph:
Answer1) Equation of the perpendicular line:
[tex]y=-\frac{x}{7}+\frac{19}{7}[/tex]2) Equation of the parallel line:
[tex]y=7x+17[/tex]In 2009, there were 6.1 million females enrolled in degree granting institutions of higher education. over the next several years this number increased at a rate of 400,000 per year. estimate the number of females enrolled in 2024. y = ______ millionthe equation of the line that models this information is;y = 0.4t + 6.1Determine what year 12.9 million females will be enrolled.
Notice that
400,000 = 0.4 million
That's why the equation that models that information has the factor 0.4, since it expresses the result in millions of females.
Now, we need to notice that t, in the expression 0.4t + 6.1, is the number of years passed since 2009. So, in the year 2024, we have:
t = 2024 - 2009 = 15
Therefore, the number of females enrolled in 2024 can be estimated to be:
y = (0.4 * 15 + 6.1) million
y = (6 + 6.1) million
y = 12.1 million
Now, to determine the year when 12.9 million females will be enrolled, we first need to find t corresponding to y = 12.9, and then add it to the year 2009.
y = 0.4t + 6.1
12.9 = 0.4t + 6.1
12.9 - 6.1 = 0.4t
6.8 = 0.4t
t = 6.8/0.4
t = 68/4
t = 17
Therefore, the year when it happens will be:
2009 + 17 = 2026
Determine the value of b for which x = 1 is a solution of the equation shown.
2x + 14 = 10x + b
B=
The linear equation has the solution x = 1 only if the value of b is 6
For which value of b is x = 1 a solution?
Here we have the linear equation:
2x + 14 = 10x + b
If we replace x by 1 in that equation, we will get:
2*1 + 14 = 10*1 + b
2 + 14 = 10 + b
16 = 10 + b
To find the value of b such that x = 1 is a solution, we need to isolate b, to do so we need to subtract 10 in both sides.
16 - 10 = 10 + b - 10
6 = b
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