Weare given the following quadratic equation, and asked to find all its real solutions:
9 z^2 - 30 z + 26 = 1
we subtract "1" from both sides in order to be able to use the quadratic formula if needed:
9 z^2 - 30 z + 26 - 1 = 0
9 z^2 - 30 z + 25 = 0
we notice that the first term is a perfect square:
9 z^2 = (3 z)^2
and that the last term is also a perfect square:
25 = 5^2
then we suspect that we are in the presence of the perfect square of a binomial of the form:
(3 z - 5)^2 = (3z)^2 - 2 * 15 z + 5^2 = 9 z^2 - 30 z + 25
which corroborates the factorization of the trinomial we had.
Then we have:
(3 z - 5)^2 = 0
and the only way such square gives zero, is if the binomial (3 z - 5) is zero itself, which means:
3 z - 5 = 0 then 3 z = 5 and solving for z: z = 5/ 3
Then the only real solution for this equation is the value:
z = 5/3
we have to write the equation of the line using y=mx+b passing through the points (6,2) and (2,4)
The general equation of line passes through point (x_1,y_1) and (x_2,y_2) is,
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]Determine the equation of line passes through point (6,2) and (2,4).
[tex]\begin{gathered} y-2=\frac{4-2}{2-6}(x-6) \\ y-2=\frac{2}{-4}(x-6) \\ y=-\frac{1}{2}x+3+2 \\ =-\frac{1}{2}x+5 \end{gathered}[/tex]So equation of line is y = -1/2x + 5.
How many pounds of candy that sells for $0.87 per lb must be mixed with candy that sells for $1.22 per lb to obtain 9 lb of a mixture that should sell for $0.91 per lb?$0.87-per-lb candy: _____lb$1.22-per-lb candy: _____lb(Type an integer or decimal rounded to two decimal places as needed.)
Let x and y be the candy pounds that sells for $0.87 and $1.22 , respectively. Since they both must add up to 9 lb, we have
[tex]x+y=9...(A)[/tex]On the other hand, the mixture should sell for $0.91 per lib, so we can write,
[tex]0.87x+1.22y=9\times0.91[/tex]Or euivalently,
[tex]\begin{gathered} \frac{0.87}{0.91}x+\frac{1.22}{0.91}y=9 \\ that\text{ is, } \\ 0.95604x+1.340659y=9...(B) \end{gathered}[/tex]Then, we need to solve the following system of equations:
[tex]\begin{gathered} x+y=9...(A) \\ 0.95604x+1.340659y=9 \end{gathered}[/tex]Solving by elimination method.
In order to eliminate variable x, we can to multiply equation (A) by -0.95604 and get an equivalent system of equations:
[tex]\begin{gathered} -0.95604x-0.95604y=-8.60439 \\ 0.95604x+1.340659y=9 \end{gathered}[/tex]Then, by adding both equations, we get
[tex]0.384619y=0.39561[/tex]Then, y is given by
[tex]\begin{gathered} y=\frac{0.39561}{0.384619} \\ y=1.02857 \end{gathered}[/tex]Once we have obtained the result for y, we can substitute in into equation (A), that is,
[tex]x+1.02857=9[/tex]then, x is given as
[tex]\begin{gathered} x=9-1.02857 \\ x=7.9714 \end{gathered}[/tex]Therefore, by rounding to two decimal places, the answer is:
$ 0.87 per lb of candy: 7.97 lb
$1.22-per-lb of candy: 1.03 lb
Which equation represents the line that is perpendicular to y = 3/4x+ 1 and passes through (-5,11)A.y=-4/3x+13/3B.y=-4/3x+29/3C.y=3/4x+59/4D.y=3/4x-53/4
Answer:
A.y=-4/3x+13/3
Step-by-step explanation:
The equation of a line has the following format:
y = ax + b
In which a is the slope.
Perpendicular to y = 3/4x+ 1:
Two lines are perpendicular if the multiplication of their slopes is -1.
Here the slope is 3/4.
In the answer to this exercise, the slope is a.
So
[tex]\frac{3}{4}\ast a=-1[/tex][tex]\frac{3a}{4}=-1[/tex]Now, cross multiplication
3a = -4
a = -4/3
So, for now, the equation is:
y = (-4/3)x + b
Passes through (-5,11):
This means that when x = -5, y = 11. So
11 = (-4/3)*(-5) + b
11 = (20/3) + b
b = 11 - (20/3)
[tex]11-\frac{20}{3}=\frac{\frac{3}{1}\ast11-\frac{3}{3}\ast20}{3}=\frac{33-20}{3}=\frac{13}{3}[/tex]So the correct answer is:
A.y=-4/3x+13/3
Order each set of numbers from least to greatest. numbers are in the photo
Answer:
Explanation:
Given the below set of numbers;
[tex]\lbrace2.8,-2\frac{3}{4},3\frac{1}{8},-\bar{2.2}\rbrace[/tex]Let's go ahead and reduce the mixed fractions to improper fractions;
[tex]undefined[/tex]Translate the sentence into an inequality.Twice the difference of a number and 2 is at least - 29.Use the variable y for the unknown number.
We will start translating the phrase "the difference of a number and 2", using y as the unknown number:
[tex]y-2[/tex]Next, using that expression, we translate the phrase "twice the difference of a number and 2", in this step, we multiply the whole previous expression by 2:
[tex]2(y-2)[/tex]To continue we consider the phrase "is at least -29", this means that the previous expression 2(y-2) has to be at least -29 or it can be greater than -29. This is represented in the following expression:
[tex]2(y-2)\ge-29[/tex]Where the symbol ≥ means greater or equal to.
Answer:
[tex]2(y-2)\ge-29[/tex]what is 6 / 1/5 helppppp
We have the following:
[tex]6\div\frac{1}{5}[/tex]we carry out the operation, in a division, a cross multiplication is made, like this
[tex]\frac{6}{1}\div\frac{1}{5}=\frac{30}{1}=30[/tex]Therefore, the answer is 30
7. (-/5 Points]DETAILSMY NOTESASMaurice is traveling to Mexico and needs to exchange $390 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?pesos
Let's apply Rule of 3
1 dollar ---------------- 12.29 Mexican pesos
390 dollars----------- x
x= 390 . 12.29 = 4793.1 Mexican pesos
A carpenter is building a set of trusses to support the roof of a residential home. In theblueprints, she has determined that she needs to make a support triangle with an area 56 m². She knows that the base must be 1 less than 2 times the height. Write the equation thatcorrectly shows the area of the triangle in terms of its height, h.
We are told that we want a triangle of area 56. Recall that the area of a triangle of base b and height h is given by the formula
[tex]\frac{b\cdot h}{2}[/tex]In our case we want
[tex]\frac{b\cdot h}{2}=56[/tex]now, we want to find an expression for b. We are told that the base is one less than twice the height. That is, we take the height, multiply it by 2, and then subtract 1. That would lead to
[tex]b=2h\text{ -1}[/tex]so we have
[tex]\frac{h(2h\text{ -1\rparen}}{2}=56[/tex]so the second option is correct.
Melody has a monthly commission plan under which she receives 2% on the first $40,000 ofsales during the month and 3% on sales above $40,000. If Melody has sales of $73,000 during amonth, computer her commission for that month.
Melody sales is above $40, 000
For sales above $40, 000, she gets 3%
So Melody commission for that month is 3% of $73,000
= 3/100 x $73000
= 3 x $730
=$2190
Find the X intercept and coordinate of the vertex for the parabola Y=X^2+ 4X -21 ,if there is more than one Y intercept separate them with commas.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
y = x² + 4x - 21
Step 02:
parabola equation:
y = x² + 4x - 21
a = 1
b = 4
c = -21
x-intercepts:
x² + 4x - 21 = 0
(x + 7)(x - 3) = 0
x1 = - 7
x2 = 3
(-7 , 0)
(3 , 0)
vertex:
[tex]xv\text{ = }\frac{-b}{2a}=\frac{-4}{2\cdot1}=-2[/tex][tex]\begin{gathered} yv=xv^2+4xv\text{ -21} \\ yv=(-2)^2+4(-2)\text{ - 21 = - 25} \end{gathered}[/tex](xv , yv)
(- 2, -25)
The answer is:
x-intercepts:
(-7 , 0)
(3 , 0)
vertex:
(- 2, -25)
The drawing below represents the frame for an isosceles triangle-shaped roof. The height of the roof is 4 feet. What is the distance from Point A to Point B in feet? B 41 3 feet 8v 6 feet 8V3 feet 8 feet
We can make a drawing to see better:
In the picture above, we can see the sides AC and BC are equals because triangle ABC is isosceles, and also the segments AD and DB are equals for the same reason.
We can calculate the lenght of segment AD as:
[tex]\begin{gathered} \tan (30)=\frac{CD}{AD} \\ AD=\frac{CD}{\tan (30)}=\frac{4}{\frac{1}{\sqrt[]{3}}} \\ AD=4\cdot\sqrt[]{3} \end{gathered}[/tex]With the lenght of segment AD we can calculate the lenght of AB as:
[tex]AB=2\cdot AD=8\cdot\sqrt[]{3}[/tex]The correct answer is in yellow.
find the average rates of change of each function for each 1-hour interval from t=0 to t=6
Step 1
Given;
[tex]Organization\text{ A, B and C}[/tex]Required; To determine the type of function representing each company
Step 2
Determine the type of function for company A
[tex]Since\text{ the number of donations quadruples each hour company A has an exponential}[/tex]An exponential function represents the number of donations collected by organization A
Determine the type of function for company B
[tex]Rate\text{ of change=}\frac{8-4}{2-1}=\frac{12-8}{3-2}=\frac{4}{1}=4[/tex]Since the rise and the run measured at points on the table are the same a linear function represents the number of donations collected by organization B.
Determine the type of function for company C
A quadratic function represents the number of donations collected by organization C
Step 3
Fill the chart.
Average rate of change for company A
[tex]undefined[/tex]To find the angle of rotation θ between the previous x and y axes and the new x′ and y′ axes, we use the formula: cotangent of 2 theta equals A minus C all over B , where 2θ lies on the interval (0°, 180°).TrueFalse
The equation is the given one:
[tex]\cot 2\theta=\frac{A-C}{B}[/tex]Which is the same as:
[tex]\tan 2\theta=\frac{B}{A-C}[/tex]And the interval of the angle θ must be (0°,90°) so, 2θ must be (0°,180°).
So the statement is True.
A Place from this table is chosen at random. Let event A= The place is a city. what is P(Ac)?
Okay, here we have this:
How The probability of an event is a ratio that compares the number of calculating favorable outcomes with the number of possible outcomes. We obtain:
[tex]P(A^C)=\frac{numberof\text{cities}}{numberofplaces}=\frac{4}{7}[/tex]Finally we obtain that the correct answer is the option B.
Which function is undefined for x = 0?O y=³√x-2Oy=√x-2O y=³√x+2Oy=√x+2
The above function is defined for (x=0)
From the question, we have
Function 1 - y = ∛x-2
Function 2 - y = √x-2
Function 3 - y = ∛x+2
Function 4 - y = √x+2
substituting (x = 0) to determine which function is undefined for (x = 0).
Function 1 - y = ∛x-2
substituting (x = 0), we get
y=∛-2
The above function is defined for (x=0).
Function 2 - y = √x-2
substituting (x = 0), we get
y = √-2
The above function is defined for (x=0).
Function 3 - y = ∛x+2
substituting (x = 0), we get
y = ∛2
The above function is defined for (x=0).
Function 4 - y = √x+2
substituting (x = 0), we get
y = √2
The above function is defined for (x=0).
Hence, it can be concluded that the above function is defined for (x=0)
Subtraction:
Subtraction represents the operation of removing objects from a collection. The minus sign signifies subtraction −. For example, there are nine oranges arranged as a stack (as shown in the above figure), out of which four oranges are transferred to a basket, then there will be 9 – 4 oranges left in the stack, i.e. five oranges. Therefore, the difference between 9 and 4 is 5, i.e., 9 − 4 = 5. Subtraction is not only applied to natural numbers but also can be incorporated for different types of numbers.
The letter "-" stands for subtraction. Minuend, subtrahend, and difference are the three numerical components that make up the subtraction operation. A minuend is the first number in a subtraction process and is the number from which we subtract another integer in a subtraction phrase.
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Please Help! I will give brainliest! Need the correct answer!
Answer:
HL
Step-by-step explanation:
The two triangles are rectangle and have a congruent hypotenuse and a congruent leg
how many terms are there in each of the following sequences?:
a) The given sequence is expressed as
52, 53, 54, 55, .......252
The first step is to determine the type of sequence by comparing the consecutive terms. We can see that there is a common difference, d between the consecutive terms.
d = 53 - 52 = 54 - 53 = 1
This means that it is an arithmetic sequence. The formula for determining the nth term of an arithmetic sequence is expressed as
an = a1 + (n - 1)d
where
an is the nth term of the sequence
n is the number of terms in the sequence
d is the common difference
a1 is the first term
From the information given,
a1 = 52, d = 1, an = 252
thus, we have
252 = 52 + (n - 1)1
252 = 52 + n - 1
252 = 52 - 1 + n = 51 + n
n = 252 - 51
n = 201
There are 201 terms in the sequence
g(t)=t^2 - 2f(t) = 4t+4Find g(t)/f(t)
Generate ordered pairs for y = x^2 - 9 using x = -4,-2,0,2 and 4. Identify the corresponding graph.
Given,
The equation is y=x^2-9.
The values of x are -4, -2, 0, 2, 4.
The odered pair are,
At x= - 4,
The value of y is,
[tex]y=(-4)^2-9=7[/tex]The ordered pair is (-4, 7).
At x= - 2,
The value of y is,
[tex]y=(-2)^2-9=-5[/tex]The ordered pair is (-2, -5).
At x= 0,
The value of y is,
[tex]y=(0)^2-9=-9[/tex]The ordered pair is (0, -9).
At x= 2,
The value of y is,
[tex]y=(2)^2-9=-5[/tex]The ordered pair is (2, -5).
At x= 4,
The value of y is,
[tex]y=(4)^2-9=7[/tex]The ordered pair is (4, 7).
The corresponding graph is,
Question 3 of 8
What is the length of CD?
B
15- X
с хр
5
E
20
1
Answer here
Explanation
Triangles ABC and CDE are congruent, then:
[tex]\begin{gathered} \frac{15-x}{20}=\frac{x}{5} \\ 5(15-x)=20x \\ 75-5x=20x \\ 75=20x+5x \\ 75=25x \\ \frac{75}{25}=x \\ 3=x \end{gathered}[/tex]Answer
x=3
it wants me to solve for the other leg and for the hypotenuse of the 45-45-90 triangle
Given:
In the given 45-45-90 triangle,
Use the tan ratio,
[tex]\begin{gathered} \tan 45^{\circ}=\frac{opposite\text{ side}}{\text{adjacent side}} \\ \tan 45^{\circ}=\frac{v}{7} \\ 1=\frac{v}{7} \\ v=7 \end{gathered}[/tex]Use the cosine ratio,
[tex]\begin{gathered} \cos 45^{\circ}=\frac{adjacent\text{ side}}{\text{hypotenuse}} \\ \cos 45^{\circ}=\frac{7}{u} \\ \frac{1}{\sqrt[]{2}}=\frac{7}{u} \\ u=\frac{7}{\sqrt[]{2}} \\ u=7\sqrt[]{2} \end{gathered}[/tex]Answer: option a)
[tex]u=7\sqrt[]{2},v=7[/tex]what is the value of the exponents of x in the simplify expression?
Let's use the following property:
[tex]x^y\cdot x^z=x^{y+z}[/tex][tex](x^{-3}y^5z^{-4})\cdot(x^6y^{-7}z^{-2})=x^{-3+6}y^{5-7}z^{-4-2}=x^3y^{-2}z^{-6}[/tex]Instructions: Use the given information to answer the questions and interpret key features. Use any method of graphing or solving. Round to one decimal place, if necessary.
SOLUTION
The given equation is:
[tex]h(x)=-x^2+10x+9.5[/tex]The graph of the function is shown.
The irrigation system is positioned 9.5 feet above the ground to start
The spray reaches maximum height of 34.5 feet at a horizontal distance of 5 feet away from the sprinkler head.
The spray reaches all the way to the ground about 10.874 feet away.
Solve. Write in scientific notation. 1.44 x 108 1.2 x 105
1.44 x 1081.2 x 105 =
4x - 4y = 20y = -5Solve the system of linear equations by graphing
The given system of equation:
4x - 4y = 20
y = -5
Plot the equation in the graph
the point at which both the lines meet is the solution of the system of equation:
the graph is :
The lines intersect at (0,-5) i.e x = 0, y = -5
So, the solution of the system of equation is x = 0 & y = -5
Answer:
x = 0
y = -5
Gerardo is skiing on a circular ski trail that has a radius of 0.9 km. Gerardo starts at the 3-o'clock position and travels 2.6 km in the counter-clockwise direction.How many radians does Gerardo sweep out? ______radians When Gerardo stops skiing, how many km is Gerardo to the right of the center of the ski trail?______ km When Gerardo stops skiing, how many km is Gerardo above of the center of the ski trail? ____km
Circle and Angles
Gerardo is skiing on a trail that has a radius of r = 0.9 km
He starts skiing at the 3-o'clock position. This means he is initially at the right of the center of the circular trail. This position corresponds to the zero degrees (or radians) reference.
The arc length of a circle of radius r is given by:
[tex]L=\theta r[/tex]Where θ is the central angle in radians.
We know Gerardo travels L=2.6 km in the counter-clockwise direction, thus the angle is calculated by solving for θ:
[tex]\theta=\frac{L}{r}\text{ }[/tex]Substituting:
[tex]\theta=\frac{2.6}{0.9}=2.8889rad\text{ }[/tex]Gerardo swept out 2.8889 radians.
Now we need to calculate the rectangular coordinates of the final position where Gerardo stopped skiing. Since the angle is less than one turn of the trail, and the angle is measured counter-clockwise, we can use the formulas:
x = r cos θ
y = r sin θ
Substituting:
x = 0.9 cos 2.8889 rad
x = -0.87 km
y = 0.9 sin 2.8889 rad
y = 0.23 km
Gerardo is -0.87 km to the right of the center. In fact, he is 0.87 km to the left of the center.
Gerardo is 0.23 km above the center of the ski trail.
The school population for a certain school is predicted to increase by 50 students per year for the next 14 years. If the current enrollment is 600 students, what will the enrollment be after 14 years?
Suppose that the school population is 600 students, and that it is predicted to increase by 50 students per year for the next 14 years. Thus we will have:
[tex]\begin{gathered} \text{Year 0: }600 \\ \text{Year 1: }600+50=600+1\cdot50 \\ \text{Year 2: }600+50+50=600+2\cdot50 \\ \ldots \\ \text{Year 14: }600+\underbrace{50+\cdots+50}=600+50\cdot14=600+700=1300 \end{gathered}[/tex]This means that the enrollment after 14 years will be of 1300 students in total.
A penny-farthing is a bicycle with a very large front wheel and a much smaller back wheel. Penny-farthings were popular in the 1800s and were available in different sizes. The ratio of the diameter of the front wheel of a penny-farthing to the diameter of the back wheel is 13:4. What is the ratio of the circumference of the front wheel to the circumference of the back wheel? Explain.
The Solution:
It is given in the question that the ratio of the diameter of the front wheel of a penny-farthing to the diameter of the back wheel is 13:4
[tex]\begin{gathered} \frac{D}{d}=\frac{13}{4} \\ \text{Where} \\ D=\text{diameter of the front wheel} \\ d=\text{diameter of the back wheel} \end{gathered}[/tex]We are required to find the ratio of the circumference of the front wheel to the circumference of the back wheel.
Step 1:
The formula for the circumference of a wheel (that is, a circle) is
[tex]\text{ circumference of a wheel = 2}\pi r=\pi d[/tex]Step 2:
We shall find the ratio of the circumference of the front wheel to the circumference of the back wheel.
[tex]\begin{gathered} \frac{\pi D}{\pi d}=\frac{13\pi}{4\pi}=\frac{13}{4} \\ \text{ So,} \\ 13\colon4 \end{gathered}[/tex]Therefore, the required ratio is 13:4
an entry commercial break 3.6 minutes if each commercial takes 0.6 minutes ,how many commercials will beplayed
We have
an entry commercial break 3.6 minutes
each commercial takes 0.6 minutes
Then we need to divide 3.6 between 0.6 in order to know how many commercials
[tex]\frac{3.6}{0.6}=6\text{ }[/tex]6 commercials in 3.6 minutes
Select the correct answer. A pyramid is placed inside a rectangular prism with height h. Area of the base of the pyramid and the prism is B. A pyramid is placed inside a prism as shown. The pyramid has the same base area, B, as the prism but half the height, h, of the prism. Which expression gives the volume of the pyramid? A. V = 2 3 B h B. V = 1 4 B h C. V = 2 B h D.
An expression which gives the volume of this pyramid is D. V = 1/6 Bh.
How to calculate the volume of a pyramid?Mathematically, the volume of a pyramid can be calculated by using this formula:
V = 1/3 × B × h
Where:
V represents the volume of a pyramid.h represents the height of a pyramid.B represents the base area of a pyramid.Since the pyramid has the same base area (B) as the prism but half (1/2) the height (h) of the prism, the new volume of this pyramid can be derived as follows:
New height, h = h/2
Substituting the parameters into the formula, we have:
Volume, V = 1/3 × B × (h/2)
Volume, V = 1/6 Bh
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Complete Question:
A pyramid is placed inside a prism as shown. The pyramid has the same base area, B, as the prism but half the height, h, of the prism. Which expression gives the volume of the pyramid?
A. V = 2/3 Bh
B. V = 1/4 Bh
C. V = 2 Bh
D. V = 1/6 Bh