The quadratic function forms a parabola. The vertex form of the equation is expressed as
y = a(x - h)^2 + k
Where
h and k are the x and y coordinates of the parabola's vertex. Given that the vertex is (4, 5),
h = 4, k = 5
Substituting these values into the above equation, it becomes
y = a(x - 4)^2 + 5
Given that the parabola passes through the point, (1, 2), we would substitute x = 1 and y = 2 into y = a(x - 4)^2 + 5. It becomes
2 = a(1 - 4)^2 + 5
2 = a * 9 + 5
2 = 9a + 5
9a = 2 -5
9a = - 3
a = - 3/9 = - 1/3
Substituting a = - 1/3 into y = a(x - 4)^2 + 5, the equation would be
[tex]y\text{ = -}\frac{1}{3}(x-4)^2\text{ + 5}[/tex]Write expression in terms of sine and cosine, and simplify so that no quotients appear in final expression.
Use the identity
[tex]1+\tan ^2x=\sec ^2x[/tex]Then the expression becomes
[tex]\frac{\sec^2x}{\sec x}=\sec x[/tex]Now, sec x is the reciprocal of cos x.
[tex]\frac{1+\tan^2x}{\sec x}=\frac{1}{\cos x}[/tex]Solve the equation using the Complete The Square Methodx^2+12x=13Do your work on paper. Take a picture, and upload it here. Show all your work/steps.
Given the quadratic equation:
[tex]x^2+12x=13[/tex]We can rewrite the equation as follows:
[tex]\begin{gathered} x^2+12x-13=0 \\ x^2+2\cdot6\cdot x-13=0 \\ x^2+2\cdot6\cdot x+6^2-6^2-13=0 \\ (x^2+2\cdot6\cdot x+6^2)-36-13=0 \end{gathered}[/tex]We see that the term inside the parenthesis is a perfect square polynomial. Then:
[tex](x+6)^2-49=0[/tex]Solving for x:
[tex](x+6)^2=49[/tex]Taking the square on both sides:
[tex]\begin{gathered} \sqrt{(x+6)^2}=\sqrt{49} \\ \\ |x+6|=7 \end{gathered}[/tex]This equation can turn into two equations:
[tex]\begin{gathered} x+6=7...(1) \\ x+6=-7...(2) \end{gathered}[/tex]Solving (1):
[tex]\begin{gathered} x=7-6 \\ \\ \Rightarrow x=1 \end{gathered}[/tex]Solving (2):
[tex]\begin{gathered} x=-7-6 \\ \\ \Rightarrow x=-13 \end{gathered}[/tex]Finally, the solutions to the equation are:
[tex]\begin{gathered} x_1=1 \\ \\ x_2=-13 \end{gathered}[/tex]what is the value of x in the equation 2.5 - 0.25x - -3
The given equation is expressed as
2.5 - 0.25x = -3
Subtracting 2.5 from both sides of the equation, we have
2.5 - 2.5 - 0.25x = -3 - 2.5
- 0.25x = - 5.5
Dividing both sides of the equation by - 0.25, we have
- 0.25x/- 0.25 = - 5.5/- 0.25
x = 22
zero, depending on your answer.|A flashlight is projecting a triangle onto a wall, as shown below.20nThe original triangle and its projection are similar. What is the missing length n on the projection?C 19.230O13.328
If two triangles are similar, it means that the ratio of their corresponding sides is the same. By applying this to the given triangles, it means that
20/n = 16/24
By crossmultiplying, we have
n x 16 = 20 x 24
16n = 480
n = 480/16
n = 30
The second option is correct
There were 24 participants in a recent study on personality traits. After being shown five images (1, 2, 3, 4, and 5), each participant selected the one that was most appealing to them. (a) The image each participant selected appears below. Complete the frequency distribution for the data.
The given images are 1, 2, 3, 4, 5
The total number of participants = 24
The frequency of each image is the number of times it appears in the distribution
Frequency of 1 = 6
Frequency of 2 = 5
Frequency of 3 = 4
Frequency of 4 = 3
Frequency of 5 = 6
The frequency distribution table is therefore shown below:
Calculator B В What is the measure of D? 25 ft Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth. © 45 ft D mD =
Given data:
The given right angle triangle.
The expression for tan(D) is,
[tex]\tan (D)=\frac{BC}{DC}[/tex]Substitute the given values in the above expression.
[tex]\begin{gathered} \tan (D)=\frac{25\text{ ft}}{45\text{ ft}} \\ D=\tan ^{-1}(\frac{25}{45})^{} \\ =29.05^{\circ} \end{gathered}[/tex]Thus, the value of angle D is 29.05 degrees.
The value of y is directly proportional to the value of x. If y = 3 whenx = 7, what is the value of x when y= 10.5? *Hint: Now you are looking for the "X". (show work)
we have that y=k*x, therefore we can say that k=y/x
[tex]k=\frac{3}{7}[/tex]so, if we have y= 10.5 we get that
[tex]x=\frac{y}{k}=\frac{10.5}{\frac{3}{7}}=24.5[/tex]x=24.5
What is the sum of the rational expressions below?3xХ+X+9 X-4O A.4x2-3xx2 +5X-36O B. 4x2-3x2x+5C.4x2x+5hosD.4xx2 +5x - 36
Given
[tex]\frac{3x}{x+9}+\frac{x}{x-4}[/tex]To find the sum of the rational expressions.
Explanation:
It is given that,
[tex]\frac{3x}{x+9}+\frac{x}{x-4}[/tex]Then,
[tex]\begin{gathered} \frac{3x}{x+9}+\frac{x}{x-4}=\frac{3x(x-4)+x(x+9)}{(x+9)(x-4)} \\ =\frac{3x^2-12x+x^2+9x}{x^2+9x-4x-36} \\ =\frac{4x^2-3x}{x^2+5x-36} \end{gathered}[/tex]Hence, the answer is option A).
f(x)=(0.13x⁴+0.22x³)-0.88x²-0.25x-0.09for this polynomial use a graph and find the minimum and maximum values written as coordinates
The given function is:
[tex]f(x)=0.13x^4+0.22x^3_{}-0.88x^2-0.25x-0.09[/tex]The graph for the polynomial f(x) is shown below:
At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 4 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 22 feet high?
[tex]h^{\prime}=\frac{4}{1089\pi}ft\text{ /min}[/tex]
STEP - BY - STEP EXPLANATION
What to find?
dh/dt
Given that;
At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 4 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude.
Since we have that the rate is 4 cubic per minute, then dv/dt = 4 (since v is the volume of the cone at time t.)
The formula for the volume of a cone is
[tex]V=\frac{1}{3}\pi r^2h---------------(1)[/tex]Where r is the radius and h is the height.
We have that, the diameter of the cone is approximately three times the altitude.
That is;
diameter = 3h
But, d = 2r
⇒2r = 3h
⇒ r = 3h/2
Now, we substitute r=3h/2 into equation (1).
[tex]V=\frac{1}{3}\pi(\frac{3h}{2})^2h[/tex][tex]=\frac{1}{3}\pi(\frac{9h^2}{4})h[/tex][tex]V=\frac{3}{4}\pi h^3[/tex]Now, differentiate the above with respect to t.
[tex]\frac{dv}{dt}=\frac{3}{4}\pi3h^2\frac{dh}{dt}[/tex]Simplify .
[tex]\frac{dv}{dt}=\frac{9}{4}\pi h^2\frac{dh}{dt}[/tex]Make dh/dt subject of formula.
[tex]\frac{dh}{dt}=\frac{dv}{dt}\times\frac{4}{9\pi h^2}----------(2)[/tex]Recall that, dv/dt = 4 and h=22
Substituting the values into equation (2), we have;
[tex]\frac{dh}{dt}=4\times\frac{4}{9\pi(22)^2}[/tex][tex]=\frac{16}{9\pi\text{ (484)}}[/tex][tex]=\frac{4}{9\pi\text{ (121)}}[/tex][tex]=\frac{4}{1089\pi}[/tex]Therefore, h' = dh/dt = 4/1089π ft/min.
x2 - 4x + 4 = 0 following quadratic equations graphically
Since we have to solve it graphically, we need to plot the function:
Here, we can see that the point where the graph touches the y axis is when x = 2
The solution to the equation is x = 2
The seventh decile is the ___ percentile in terms of the data set.
The statement is given as
The seventh decile is the ___ percentile in terms of the data set.
ExplanationThe seventh decile is the 70-th percentile in terms of the data set.
AnswerHence the answer is 70th percentile in terms of data set.
a blueprint for a house has a scale factor of 1 inch 3 ft. a wall in the blueprint is 5 in what is the length of the actual wall in feet
15 ft
Explanation
you can easily solve this by using a rule of three
Step 1
Let x represents the actual length of the wall in feet
is
[tex]1\text{ inc}\rightarrow3\text{ ft}[/tex]then
[tex]5\text{ in}\rightarrow x[/tex]as the proportion is the same
[tex]\begin{gathered} \frac{1}{3}=\frac{5}{x} \\ \text{cross multiply} \\ x\cdot1=5\cdot3 \\ x=15\text{ ft} \end{gathered}[/tex]so, the answer is 15 ft
I hope this helps you
Calculate the simple interest due on a 54-day loan of $3600 if the interest rate is 3%. (Round your answer to the nearest cent.)
Answer:
$5832.00cents
Step-by-step explanation:
PRT÷100
$3600 ×54 ×3 /100
= $5832.00cents.
Find the equation of a line passing through the points (3,-4) and (1,2)
Explanation:
The equation of a line in slope-intercept form is:
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
The formula for the slope of a line with points (x1, y1) and (x2, y2) is:
[tex]m=\frac{y_1-y_2}{x_1-x_2}[/tex]In this case the slope is:
[tex]m=\frac{-4-2}{3-1}=\frac{-6}{2}=-3[/tex]For now we have:
[tex]y=-3x+b[/tex]To find the y-intercept b, we have to replace (x,y) for one of the given points and solve for b. If we use point (1,2):
[tex]\begin{gathered} y=-3x+b \\ \text{ replacing x = 1 and y = 2} \\ 2=-3\cdot1+b \\ 2=-3+b \\ b=2+3=5 \end{gathered}[/tex]Answer:
The equation of the line is: y = -3x + 5
Answer:
y=\neg;[3]x+5
Step-by-step explanation:
9. In April, Community Hospital reported 923 discharge days for adults and children and 107 discharge daysfor newborns. During the month, 192 adults and children and 37 newborns were discharged. Calculate theALOS for adults and children for the month of April. Round to one decimal place.
Solution:
Average Length of Stay(ALOS) is calculated by summing the number of days for all stays (where partial days, including non-overnight stays, are rounded up to the next full day) and dividing by the number of patients.
[tex]\text{alos}=\frac{total\text{ number of days}}{\text{totl number of patients}}[/tex]The total number of days is
[tex]\begin{gathered} =923+107 \\ =1030 \end{gathered}[/tex]The total number of patients is
[tex]\begin{gathered} =192+37 \\ =229 \end{gathered}[/tex]By applying the formula above, we will have
[tex]\begin{gathered} \text{alos}=\frac{total\text{ number of days}}{\text{total number of patients}} \\ \text{alos}=\frac{1030}{229} \\ \text{alos}=4.4978 \\ \text{alos}\approx4.5 \end{gathered}[/tex]Hence,
The final answer is =4.5
The value of a ratio is 4/3. The second quantity in the ratio is how many times the first quantity in the ratio?
The second quantity in the ratio is 3/4 times the first quantity in the ratio.
How many times in the second quantity in the ratio more than the first quantity?Ratio is the number of times that one value is contained within other value(s). This ratio is expressed as an improper fraction. A fraction is a non-integer that is made up of a numerator and a denominator. An improper fraction is a fraction in which the numerator is larger than the denominator.
In order to determine the number of times the second quantity is greater than the first quantity, determine the inverse of the given fraction.
The inverse of 4 /3 is 3/4.
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Polygon ABCD with A (2,4), B (-4,-8), C (0,4), and D (12,-2), is dilated are the new coordinates? Is this a reduction or enlargement?
For this exercise it is important to know that, in Dilations:
1. When the scale factor is greater than 1, the Image (the figure obtained after the transformation) is an enlargement.
2. When the scale factor is greater than 0 but less than 1, the Image is a reduction:
[tex]0You know that the coordinates of the vertices of the polygon ABCD are:[tex]\begin{gathered} A(2,4) \\ B(-4,-8) \\ C(0,4) \\ D(12,-2) \end{gathered}[/tex]And the scale factor is:
[tex]k=\frac{1}{2}[/tex]Since:
[tex]0<\frac{1}{2}<1[/tex]It is a reduction.
You can identify that the rule of this transformation is:
[tex]undefined[/tex]Drag numbers in the box to make each comparison true
1) Let's find the results of each inequality/equality so that we write out the right choices.
2) Let's begin with the equation. We're going to resort to Algebra:
[tex]\begin{gathered} 12\times\frac{4}{x}=8 \\ 12\times4=8x \\ 48=8x \\ 8x=48 \\ \frac{8x}{8}=\frac{48}{8} \\ x=6 \\ 12\times\frac{4}{6}=8{\color{DarkBlue} } \end{gathered}[/tex]Based on that, we can tell that and filling in the gap with other whole numbers the following:
[tex]\begin{gathered} 12\times\frac{4}{8}<8 \\ 6<8 \\ 12\times\frac{4}{6}=8 \\ 12\times\frac{4}{3}>8 \\ 16>8 \end{gathered}[/tex]Note that we picked whole numbers and it is clear that the greater the denominator the lesser the value.
there are 4 sets of balls numbered 1 through 12 placed in a bowl. if 4 balls are randomly chosen without replacement, find the probability that the balls have the same number. express your answer as a fraction
The probability is 12/194580
Explanation:The balls numbered 1 through 12 are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
If 4 balls have the same number, then there are 12 types of this arrangement:
1, 1, 1, 1
2, 2, 2, 2
3, 3, 3, 3
and so on.
There is also 4 * 12 = 48 total number of balls.
We have a permutation:
[tex]\begin{gathered} 48C4=\frac{48!}{(48-4)!4!} \\ \\ =\frac{48!}{44!4!}=194580 \end{gathered}[/tex]Finally, we
[tex]\frac{12}{194580}[/tex]This is the required probability.
Kris and Pat were born on the exact same day but not in the same year their ages are shown in the table When Pat was 30 how old was Chris
A ) 18 years old
B) 19 years old
C) 27 years old
add three to kris' age to find Pat's age
Explanation
Step 1
when Kris was 15 .how old was pat
find the rule
a) check if the difference between the ages is constant
[tex]\begin{gathered} \text{Pat's age - Kris' Age=} \\ 7-4=3 \\ 10-7=3 \\ 15-12=3 \\ y_4\text{-}15= \\ 22-x_5= \\ 26-23=3 \end{gathered}[/tex]it means Pat is 3 years older than Kriss
then
when Kris was .how old was pat
[tex]\begin{gathered} y_4\text{-}15=3 \\ y_4-15=3 \\ \text{add 15 in both sides} \\ y_4-15+15=3+15 \\ y_4=18 \end{gathered}[/tex]Step 2
when Pat was 22, How old was Kris
the difference between the ages is the same, 3
so
[tex]\begin{gathered} 22-x_1=3 \\ \text{subtract 22 in both sides} \\ 22-x_5-22=3-22 \\ -x_5=-19 \\ x_5=19 \end{gathered}[/tex]Step 3
when Pat was 30, How old was kris
replace
[tex]\begin{gathered} \text{Pat's age -Kris' age=3} \\ 30-\text{Kri's age=3} \\ \text{subtract 30 in both sides} \\ 30-\text{Kri's age-30=3-}30 \\ -\text{Kris'age =-27} \\ \text{Kris'age }=27 \end{gathered}[/tex]Step 4
wich choice best represents the rule
as we saw before,
add three to kris' age to find Pat's age
[tex]\text{Kris' age+3}=\text{Pat's age}[/tex]Rewrite x^2+9 in factored form
The factored form is, x =3i and x = -3i
Given:
The objective is to write x^2+9 in factored form.
The factored form can be written as,
[tex]\begin{gathered} x^2+9=0 \\ x^2=-9 \\ x=\sqrt[]{-9} \\ x=\pm3i \end{gathered}[/tex]Here, i stands for imaginary term due the square root of (-1).
Hence, the factored form is, x =3i and x = -3i.
Which relationships can be represented by the equation y=1/5x?
First, notice that 1/5 = 0.20. With this in mind, the examples that can be represented by the equation y = 1/5x are the following:
1.-Bananas are on sale for $0.20. Let x represent the number of bananas and y represent the total cost,in dollars
2.-For every 10 gallons of water, Jasmine adds 2 cups of Soap. Let x be the gallons of water and y the cups of soap.
Write 16 2/3% a as a decimal and b as a reduced fraction
Given the following percentage:
[tex]16\frac{2}{3}[/tex]Convert into a decimal and into a reduced fraction
16 + 2/3 =
16 + 2/3 = 0.166666 percent or 0.16 as a repeating decimal
To convert the decimal into a fraction:
[tex]\frac{(0.16\times10^2)-0}{10^2-1}=\frac{16}{99}[/tex]Fraction already reduced no need to simplify it further so....
= 16 / 99
Lana owns an office supply shop. At the beginning of each school year, she chooses two or three products to donate to the local middle school.
The table shows the school supplies that Lana has in her shop and how many of each kind she has in stock. Lana is considering different options of supplies to donate. For each option, determine the greatest number of identical boxes she could pack and the number of each supply item she could put in the boxes.
school supplies stock:
pencils-78
markers-110
notebooks-195
erasers-143
folders-330
A option 1: pencils and erasers: ____boxes with ____ pencils and ____ erasers
B option 2: notebooks and folders: ____ boxes with ____notebooks and ____ folders in each box.
C option 3: erasers, markers, and folders: ____ boxes with ____ erasers ____ markers and ____ folders in each box.
Answer: Step-by-step explanation: Lana needs to order supplies for the upcoming month. Her supplier sells pencils in boxes of 12, markers in boxes of 10, notebooks in boxes of 4, erasers in boxes of 6, and folders in boxes of 15. What is the least number of boxes of each supply item that Lana could order to have the same number of each item delivered? How many of each item will she get? E 3-2
Ted and his classmates started watering flowers at 11:36 andfinished watering all the flowers at 13:25. If they wateredflowers at a constant rate, at what time did they finish watering4/7 of the flowers? Give your answer in a 24-hour clock format,such as 19:00.Enter the answer
SOLUTION
The duration for watering the whole flower is
[tex]\begin{gathered} \text{ finishing time- starting time } \\ 13\colon25-11\colon36 \\ \end{gathered}[/tex]Which is
[tex]1\text{hours 49minutes or 109 minutes }[/tex]This means it takes 109minutes for watering the whole flowers.
The time taken for 4/7 of the flowers is
[tex]\begin{gathered} \frac{4}{7}0f\text{ 109} \\ \\ \frac{4}{7}\times109=\frac{436}{7}=62.29 \\ \\ \end{gathered}[/tex]Then it takes 62minutes 29seconds to wet 4/7 of the flowers
[tex]\begin{gathered} 62\text{minutes 29 seconds will be } \\ 1\text{hours 0.5minutes } \end{gathered}[/tex]The time they will finish watering 4/7 of the flowers will be
[tex]\begin{gathered} \text{starting time +Duration for 4/7 of the flowers } \\ 11\colon36+1;05 \\ 12\colon41 \end{gathered}[/tex]Problems 20 - 23. Analytically determine what type(s) of symmetry, if any, the graph of the equation would possess. Show your work.21) y^2 - xy = 6
Because of gthe graph I conclude that the graph is simmetric about the origin.
Find the product of these complex numbers.(8 + 6i)(-5 + 7i) =A.-82 - 86iB.-82 + 26iC.2 + 26iD.2 - 86i
Solution
Step 1:
Write the expression:
(8 + 6i)(-5 + 7i)
Step 2:
[tex]\begin{gathered} \left(8+6i\right)\left(-5+7i\right) \\ \\ 8\times(-5)\text{ + 8 }\times7i\text{ + 6i }\times(-5)\text{ + 6i }\times\text{ 7i} \\ \\ =\text{ -40 + 56i - 30i + 42i}^2 \\ \\ =\text{ -40 + 26i - 42} \\ \\ =\text{ - 82 + 26i} \end{gathered}[/tex]Final answer
B.
-82 + 26i
The probability is
(Round to four decimal places as needed.)
Points: 0 of 1
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Assume that when human resource managers are randomly selected, 43% say job applicants should follow up within
two weeks. If 7 human resource managers are randomly selected, find the probability that at least 2 of them say job
applicants should follow up within two weels.
Using the binomial distribution, it is found that there is a 0.7873= 78.73% probability that at least 2 of them say job applicants should follow up within two weeks.
For each manager, there are only two possible outcomes. Either they say job applicants should follow up within two weeks, or they do not say it. The opinion of a manager is independent of any other manager, which means that the binomial distribution is used to solve this question.
Binomial probability distribution
P(X = x) = Cₙ,ₓ.pˣ.(1-p)ⁿ⁻ˣ
Cₙ,ₓ = n!/x!
The parameters are:
n is the number of trials.
x is the number of successes.
p is the probability of a success on a single trial.
In this problem:
43% say job applicants should follow up within two weeks. If 7 human resource managers are randomly selected, n = 8
The probability that at least 2 of them say job applicants should follow up within two weeks is P(X≥2), which is given by,
P(X≥2) = 1 - P(X < 2)
In which:
P(X<2)=P(X=0)+P(X=1)
Then,
P(X=x) = Cₙ,ₓ.pˣ.(1-p)ⁿ⁻ˣ
P(X=0) = C₇,₀.(0.43)°.(0.57)⁷ = 0.0195
P(X=1) = C₇,₁.(0.43)¹.(0.57)⁶ = 0.1032
P(X<2) = P(X=0)+P(X=1)
= 0.0195+0.1032
= 0.2127
P(X≥2) = 1-P(X<2)
= 1 - 0.2127
= 0.7873
0.7873 = 78.73% probability that at least 2 of them say job applicants should follow up within two weeks.
Hence we get the probability as 78.73%.
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10. What is 7/4 as a mixed number?
Solution:
Given;
What is 7/4 as a mixed number?
7/4 as a mixed number can be expressed as
[tex]1\frac{3}{4}[/tex]Hence, the answer is 1 3/4