ANSWER
[tex]y=4x-\frac{15}{2}[/tex]EXPLANATION
Two lines are perpendicular if they have opposite reciprocal slopes. So, to find the line, first, we have to find the slope of the line tangent to the graph of y = 1/x at x = 2. To do so, we have to find the derivative of the function and evaluate it at x = 2. That is the slope of the tangent line at that point.
The derivative of the function is,
[tex]y^{\prime}=\left(\frac{1}{x}\right)^{\prime}=(x^{-1})^{\prime}=-1\cdot x^{-1-1}=-1x^{-2}=-\frac{1}{x^2}[/tex]So, the slope of the tangent line at x = 2 is,
[tex]y^{\prime}(2)=-\frac{1}{2^2}=-\frac{1}{4}[/tex]Therefore, the slope of the perpendicular line is the opposite reciprocal,
[tex]m=-\frac{1}{y^{\prime}(2)}=-\frac{1}{-\frac{1}{4}}=4[/tex]So we have that the perpendicular line we are looking for is,
[tex]y=4x+b[/tex]We have to find the y-intercept, b. We know that this perpendicular line intersects the graph at the tangency point, so with that point, we will find b.
The x-coordinate of the tangency point is x = 2, and the y-coordinate is,
[tex]y(2)=\frac{1}{2}[/tex]Substitute (x, y) with (2, 1/2) in the equation of the line to find b,
[tex]\begin{gathered} \frac{1}{2}=4\cdot2+b \\ \\ \frac{1}{2}=8+b \end{gathered}[/tex]Subtract 8 from both sides,
[tex]b=\frac{1}{2}-8=\frac{1-8\cdot2}{2}=\frac{1-16}{2}=-\frac{15}{2}[/tex]Hence, the equation of the line perpendicular to the tangent of the graph at x = 2 passing through the tangency point is,
[tex]y=4x-\frac{15}{2}[/tex]A rectangular auditorium seats 1749 people the number of seats in each row exceed the number of rows by 20 find the number of seats in each row
Let x = the number of seats
Let y = the number of rows.
Since the auditorium is rectangular and it has 1,749 people, then we can say that:
[tex]x\times y=1749[/tex]Then, if the number of seats "x" exceeds the number of rows "y" by 20, then we can say that:
[tex]\begin{gathered} seat=row+20 \\ x=y+20 \end{gathered}[/tex]Now we have two equations. To solve for x, let's use the substitution method.
1. Rewrite the equation 2 x = y + 20 into y = x - 20.
2. Replace the value of "y" in equation 1 by x - 20.
[tex]\begin{gathered} xy=1749 \\ x(x-20)=1749 \end{gathered}[/tex]2. Multiply x and x - 20.
[tex]x^2-20x=1749[/tex]3. Transfer the constant term 1749 on the left side of the equation. When transferring over the equal sign, the operation will change. From +1749, it becomes -1749.
[tex]x^2-20x-1749=0[/tex]4. To solve this quadratic equation, let's find the factors of -1749 that sums to -20.
a. 3 and -583 = -580
b. 11 and -159 = -148
c. 33 and -53 = -20
As we can see above, the factors of -1749 that sums to -20 are 33 and -53. Hence, the quadratic equation above can be factored to:
[tex](x+33)(x-53)=0[/tex]5. Equate each factor to zero and solve for x.
[tex]\begin{gathered} x+33=0 \\ x=-33 \end{gathered}[/tex][tex]\begin{gathered} x-53=0 \\ x=53 \end{gathered}[/tex]Since the value of x cannot be negative, then the value of x is 53.
Therefore, the number of seats in each row is 53. In addition, there are 33 rows in the auditorium.
choose the correct description of the graph of the compound inequality1: A number line with an open circle on 0 shading to the left and a closed circle on 2 shading to the right2: A number line with a closed circle on 0, shading to the left, and an open circle on 2 shading to the right3: A number line with a closed circle on 2, and shading in between 4: A number line with an open circle on 2, and shading in between
Answer:
Step-by-step explanation:
DUE IN 30 MINUTES PLEASE HELP!!!!
WITH WORK PLEASE
1. Write a piecewise defined rule from a graph
2. Write three equations and show all your steps to get full credit
For the given question,
The coordinates of the three lines are given with the end points.
Part a: The slope piece wise rule for defined graph is-
[2, 1) - as (1 - 1) is at the open interval.[1,5] - both ends with closed interval.[5,8] - both ends with the closed interval.Part b: calculate the three equations-
First find the slope for the three given lines;
slope (blue line);
coordinates - (0, -4) and (1, -1)
m = (-1 + 4)/(-1 - 0)
m = -3
Equation of line;
y + 4 = -3(x -0)
y = -3x - 4
slope (red line);
coordinates - (1, 4) and (5, 8)
m = (8 - 4)/(5 - 1)
m = 4/4
m = 1
Equation of line;
y - 4 = 1(x - 1)
y = x + 3
slope (green line);
coordinates - (5, 8) and (8, 8)
m = (8 - 8)/(8 - 5)
m = 0
Equation of line;
y - 8 = 0(x - 5)
y = 8
Thus, the three equation of line are found.
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how do you start square roots in mathmaticals?
Answer:
a square root is basically any number multiplied by itself to get another number. there are 2 types, perfect and non perfect, perfect is when the number being multiplied by itself is an a whole number, while the non perfect is when the number that is being multiplied by itself is a decimal or fraction
Step-by-step explanation:
For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.A. yes; y=2xB. yes; y=3xC. yes; y=4xD. no; y does not vary directly with x.
We are given a table with values of "x" and associated values of "y". Let's remember that direct variation implies that each value of "y" will be related with its corresponding value of "x" according to the following relationship:
[tex]y=kx[/tex]Where "k" is the constant of proportionality. If the table shown has direct variation the same constant should apply for each value. Let's take the first value of the table, that is, x = 3 and y = 6. Substituting we get:
[tex]6=k(3)[/tex]Dividing both sides by 3 we get:
[tex]\frac{6}{3}=k[/tex]Solving the operations:
[tex]2=k[/tex]Now we substitute in the relationship:
[tex]y=2x[/tex]Now, for the second value of "x", that is x = 6:
[tex]\begin{gathered} y=2(6) \\ y=12 \end{gathered}[/tex]Since we did not get the corresponding value of "y" in the table, this means that the constant of proportionality doesn't work for this value, and therefore, "y" does not vary directly with "x".
The time to complete a standardized exam is approximately normal with a mean of 70 minutes and a standard
deviation of 10 minutes. How much time should be given to complete the exam so that 80% of the students will complete the exam in the time given?
(A) 61.6 minutes
(B) 78.4 minutes
(C) 79.8 minutes
(D) 80.4 minutes
Please help with this question
Answer:
A is correct. √11 and √14 are both between 3 and 4.
How do I solve this, and how do I know if I need to add or subtract at the beginning
Let's solve the system of equations using elimination:
4x + 5y = 14
-4x -2y = -8
First, we need to get rid of one variable.
Sometimes we multiply them by some factors to have the same number on one variable, but in this case, we can subtract the x variable
4x +5y =14
-4x-2y = -8
----------------------
4x-4x =0, 5y -2y = 3y and 14-8=6
------------------------------
0+3y = 6
Solve the equation for y
Divide by 3 into both sides
3y/3 = 6/3
y = 2
Then, replace this variable on one equation, whichever you want:
4x + 5y = 14
4x +5(2) = 14
4x +10 = 14
and solve the equation for x
4x +10 = 14
Subtract 10 on both sides
4x +10-10 = 14-10
4x = 4
Divide by 4 into both sides
4x/4 = 4/4
x = 1
You can replace both variables to confirm the results:
4x +5y =14
4(1) +5(2) = 14
4 +10 = 14
14 = 14
---------------------------------
-4x -2y = -8
-4(1) -2(2) = -8
-4-4 = -8
-8 = -8
The result is correct, so my ordered pair is (x,y) = (1,2)
Find the sum of the first 9 terms of the following sequence. Round to the nearesthundredth if necessary.40,-16,32/5
SOLUTION
The following sequence is a geometric series and we have been provided with the formula
[tex]S_n=\frac{a_1-a^{}_1r^n}{1-r}[/tex]Here a1 is the first term = 40,
r is the common ratio = -0.4 (to get r, divide the second term by the first term)
n = number of terms = 9. Now let's solve
[tex]\begin{gathered} S_n=\frac{a_1-a^{}_1r^n}{1-r} \\ \\ S_9=\frac{40_{}-40\times(-0.4)^9}{1-(-0.4)} \\ \\ S_9=\frac{40_{}-(-0.0105)^{}}{1+0.4} \\ \\ S_9=\frac{40_{}+0.0105^{}}{1+0.4} \\ \\ S_9=\frac{40.0105^{}}{1.4} \\ \\ S_9=28.5789 \end{gathered}[/tex]The sum to the nearest hundredth becomes = 28.58
Michael bought 1/2 pound of Swiss cheese. He used 1/4 pound for
a sandwich. How much was left?
Answer:
1/4
Step-by-step explanation:
1/2 - 1/4 = 2/4 - 2/4 = 1/4
Answer the questions below.
determine the inverse function of f(x) = x² + x - 6, x> - 1/2, if it exists. Then, state its domain and range.
Since the domain is restricted to one side of the axis of symmetry of the full graph of f(x), an inverse does exist.
Letting [tex]f(x)=y[/tex],
[tex]x=y^2 +y-6\\\\x=\left(y+\frac{1}{2} \right)^2 -\frac{25}{4}\\\\x+\frac{25}{4}=\left(y+\frac{1}{2} \right)^2[/tex]
Since the domain is restricted to the right hand side of the axis of symmetry, and since the range of the inverse is the same as the domain of the original function, we take the positive square root.
[tex]\sqrt{x+\frac{25}{4}}=y+\frac{1}{2}\\\\y=f^{-1}(x)=\sqrt{x+\frac{25}{4}}-\frac{1}{2}[/tex]
The domain of the inverse is the range of the original function, which is [tex]\left[-\frac{25}{4}, \infty \right)[/tex].
The range of the inverse is the domain of the original function, which is [tex]\left[-\frac{1}{2}, \infty)[/tex],
Solve the following compound inequalities. Use both a line graph and interval notation to write each solution set.
The required value of t is (-∞, -6) U (4, ∞).
What is a linear inequality and how is it solved?
Expressions with linear inequalities compare any two values using inequality symbols like "<" and ">." The mathematical expression with unequal sides is known as an inequality in mathematics. Inequality is referred to in mathematics when a relationship results in a non-equal comparison between two expressions or two numbers. Polynomial inequality, rational inequality, and absolute value inequality are the several types of inequalities that exist in mathematics.
First, create an equation for the inequality. The given equation must be solved for one or more values. Represent all of the values that you found on the number line now. On the number line, use open circles to symbolize the excluded values. the interval, then. To determine whether the values satisfy the inequality equation, substitute any random value from the interval into the inequality equation. The solutions of the given inequality equation are intervals that fulfil the inequality equation.
Given, the linear inequalities are t + 1 < -5 and t + 1 > 5
Solving the first inequality, t + 1 < -5 ⇒ t < -5 -1 ⇒ t < -6
Again, solving second inequality, t + 1 > 5 ⇒ t > 5 - 1 ⇒ t > 4
Combining the solution of both the inequalities, we have;
the value of t can take any value in (-∞, -6) U (4, ∞).
Therefore, the required value of t is (-∞, -6) U (4, ∞).
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all you need is in the photo put only the answer Don't do step by step please answer fast pleaseeeeegkg
Substitute the values of the constant in the equation to obtain the value of x.
[tex]undefined[/tex]Complete the table given the following function:
f(x) = −2x^2 + 1
Answer:
if f(-4) = -31
then
f(-2) = -7
f(0) = 1
f(1) = -1
f(3) = -17
Step-by-step explanation:
You have to replace the x of the f(x) equation with said number in the left column. Then plug it into your calculator and your golden :]
f(x) = -2x²+1
= -2(-4)²+1
= -32+1
= -31x = -2f(x) = -2x²+1
= -2(-2)²+1
= -2(4)+1
= -8+1
= -7x = 0f(x) = -2x²+1
= -2(0)²+1
= -2(0)+1
= 0+1
= 1x = 1f(x) = -2x²+1
= -2(1)²+1
= -2(1)+1
= -2+1
= -1x = 3f(x) = -2x²+1
= -2(3)²+1
= -2(9)+1
= -18+1
= -17PLEASE HURRY
What is the value of the expression −34 − 17 − (−22)?
Answer:
-29
Step-by-step explanation:
subtracting the negative 22 from the -17 made it positive until you subtract the 34 which made it -29
Mr. Cumme has 8 3/4 yards of material for her class. he needs to use 1/3 of the material to make an example. How many yards of material will he need to make the example?
Given:
Mr. Cumme has 8 3/4 yards of material for her class.
he needs to use 1/3 of the material to make an example.
To find the number of yards of material that he will need, we will multiply 1/3 by 8 3/4 to get the answer.
[tex]\frac{1}{3}\times8\frac{3}{4}=\frac{1}{3}\times\frac{35}{4}=\frac{35}{12}=2\frac{11}{12}[/tex]So, the answer will be 2 11/12 yards.
Which expression is equivalent to (-3t - x) - (5y - 8x)?
Write the equation of the line in point-slope form that has a slope of 2/3 and a y-intercept of (0,-2)
y-2= 2/3 (x-0)
y-0= 2/3 (x-2)
y-0=2/3 (x+2)
y+2=2/3 (x-0)
Answer:
y + 2 = 2/3(x - 0)
Step-by-step explanation:
The general form of an equation of line with slope m, passing through point (x1, y1) is:
y - y1 = m(x - x1)
substituting values:
y - (-2) = 2/3(x - 0)
simplify
y + 2 = 2/3(x - 0)
Answer:
where
m(Gradient or slope) =2/3
y1=-2
x1=0
Match the equivalent fractions
5
13
HELPPPPPPPPP PLEASEEEEEEEEEE
Solve for b, then solve for each angle in the triangle
We know that the sum of the interior angles of a triangle equals 180, then,in this case we have the following equation:
[tex]b+2b+(b+16)=180[/tex]then, solving for b, we get:
[tex]\begin{gathered} b+2b+(b+16)=180 \\ \Rightarrow4b+16=180 \\ \Rightarrow4b=180-16=164 \\ \Rightarrow b=\frac{164}{4}=41 \\ b=41 \end{gathered}[/tex]now that we have that b = 41, we can find the measure of each angle:
[tex]\begin{gathered} b=41 \\ 2b=2(41)=82 \\ b+16=41+16=57 \end{gathered}[/tex]which set could not represent the lengths of the slides of a triangle?
hello
to solve this question, we asume it's a right angled triangle and we use pythagorean theorem
pythagorean thorem states that the square of the longest side which is the hypothenus of a triangles is equal to the sum of the square of the two other sides which are the opposite and adjacent
according to pythagorean
[tex]x^2=y^2+z^2[/tex]now we go through the options
from the first option (3, 4, 5)
the longest side here is 5
let's check if this is true
[tex]\begin{gathered} x^2=y^2+z^2 \\ 5^2=4^2+3^2 \\ 25=16+9 \\ 25=25 \end{gathered}[/tex]option 1 is represents a triangle
let's check option 2
(2, 5, 9)
longest side = 9
[tex]\begin{gathered} 9^2=2^2+5^2 \\ 81=4+25 \\ 81\ne29 \end{gathered}[/tex]option 2 does not represents a triangle and it's the right option here
What’s the correct answer answer asap for brainlist
Answer:
the answer is C..........
The triangle is translated. B' is the translated position of B.
Draw the new triangle, then verify that they are congruent with the distance formula.
The new triangle A'B'C' shown in the attached graph has the coordinates of B' are (2, 8), A' are (1, 6), and C' (4, 6).
What are congruent triangles?Two triangles are said to be congruent if their corresponding sides and angles are equal.
The given triangle ABC shown in the graph has:
The coordinates of B are (-7, 6),
The coordinates of A are (-8,4),
The coordinates of C are (-5, 4)
If B' is the translated position of B.
The new triangle A'B'C' shown in the attached graph has:
The coordinates of B' are (2, 8)
The coordinates of A' are (1, 6)
The coordinates of C' are (4, 6)
Length of AB = A'B' = √5
Length of BC = B'C' = 2√2
Length of CA = C'A' = 3
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Compute the complement rate where the trade discount rate is 15%.
The Complement rate where Trade discount rate is 15% = 100 - 15 = 85%.
Thanks
A country's population in 1993 was 204 million.In 2000 it was 208 million. Estimatethe population in 2015 using the exponentialgrowth formula. Round your answer to thenearest million.burP = AektEnter the correct answer.DONE?DOO
Here, we want to use the exponential growth formula to estimate the 2015 population
In order to use the exponential growth formula, we have to get some of the needed points like the growth rate or constant represented by K
At any point in time, A represents the initial popultaion, which is the population in 1993, given as 204 million
P represents the population at that point in time
k is what we want to calculate
t is the difference in the number of years
Thus, for between 1993 and 2015, we have ;
[tex]undefined[/tex]A bell rings every 20 minutes. A horn blows every 30 minutes. If Htel Htel heard the two sounds at 8:00AM, at what time will be hear the sounds together again.
Answer:
Htel Htel will hear the sounds again at 9:00 AM
if EB=7, find the value of CD
Both lines AB and CD are secant lines beacuse in two points they are touching the circunference. There is a Theorem which says the following
[tex]PB\cdot PA=PD\cdot PC[/tex]Since the distance from the center of the circle to each secant line is the same (5 units), we could assume that the both secant lines are similar. saying:
[tex]PB=PD;\text{ }PA=PC\Rightarrow AB=CD[/tex]Then the lenght of CD is:
[tex]CD=2\cdot EB=2(7)=14[/tex]find a solution of the quadratic equation.6x squared + 7x + 2 =0
Given the equation:
[tex]6x^2+7x+2=0[/tex]We will use the following rule to find the solution to the equation:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]From the given equation: a = 6, b = 7, c = 2
So,
[tex]\begin{gathered} x=\frac{-7\pm\sqrt[]{7^2-4\cdot6\cdot2}}{2\cdot6}=\frac{-7\pm\sqrt[]{1}}{12}=\frac{-7\pm1}{12} \\ x=\frac{-7-1}{12}=-\frac{8}{12}=-\frac{2}{3} \\ or,x=\frac{-7+1}{12}=-\frac{6}{12}=-\frac{1}{2} \end{gathered}[/tex]So, the answer will be option B) x = -1/2, -2/3