first we need to solve X
[tex]\begin{gathered} -y^2+14y-57=4x \\ x=-\frac{1}{4}y^2+\frac{7}{2}y-\frac{57}{4} \\ \end{gathered}[/tex]we need to write the equation on this form
[tex]x=a(y-h)^2+k[/tex]where h=-(b/2a) and k=c- a (b/2a)2
we obtain a,b and c from the equation to solve x
so a=-1/4, b=7/2 and c=-57/4
now lets find h and k
[tex]\begin{gathered} h=-(\frac{b}{2a}) \\ h=-(\frac{\frac{7}{2}}{2\cdot-\frac{1}{4}}) \\ \\ h=-(\frac{\frac{7}{2}}{\frac{-1}{2}}) \\ \\ h=-(-7) \\ h=7 \end{gathered}[/tex][tex]\begin{gathered} k=c-a(\frac{b}{2a})^2 \\ \\ k=-\frac{57}{4}-(-\frac{1}{4})(\frac{\frac{7}{2}}{2\cdot-\frac{1}{4}})^2 \\ \\ k=-\frac{57}{4}+\frac{1}{4}(-7)^2 \\ \\ k=-\frac{57}{4}+\frac{1}{4}(49) \\ \\ k=-\frac{8}{4} \\ k=-2 \end{gathered}[/tex]now replace a, h and k on the equation
[tex]\begin{gathered} x=a(y-h)^2+k \\ \\ x=-\frac{1}{4}(y-7)^2-2 \end{gathered}[/tex]the evrtex is (h,k)=(7,-2)
You randomly draw a marble from a bag of marbles that contains 7 blue marbles 2 green marbles and 1 red marbles
Given the following:
7 blue marbles
2 green marbles
1 red marbles
We to find the probability of not drawing a blue marble.
We will be solving it in two ways.
First let's get the total marbles
Total Marble = 7 + 2 + 1 = 10
recall that probablity is number of favourable outcome divide by number of total outcome.
So,
probablity of Drawing a Blue Marble is = 7/10
probability of not Drawing Blue Marbles = 1 - Probability of Drawing Blue Marbles
= 1 - 7/10
= 10 - 7
10
= 3/10
OR
Probability of not Drawing Blue Marbles = Probablity of drawing Green or Red Marbles.
= 2/10 + 1/10
= 3/10
Therefore, the probability of not Drawing Blue Marbles is 3/10.
Gabe made a scale drawing of a neighborhood park. The scale of the drawing was 1 millimeter : 6 meters. If the actual length of the volleyball court is 18 meters, how long is the volleyball court in the drawing?
Write the sentence as an equation. 136 is equal to 194 times b Type a slash (/) if you want to use a division sign.
Given the sentence 136 is equal to 194 times b, we are to write this statement as an equation.
Let us take it one after the other.
For 194 times b, this can be written as;
= 194 * b
= 194b .
Since 136 is equal to the exxpression, the final equation will be gotten by simply equationf 136 to 194b as shown;
136 = 194b
You can then re-arrange
194b = 136
Hence the reuired equation is 194b = 136
What is 10/12 written in simplest form?
ANSWER:
[tex]\frac{5}{6}[/tex]STEP-BY-STEP EXPLANATION:
We have the following fraction
[tex]\frac{10}{12}[/tex]Now to reduce to its simplest form, we must simplify
[tex]\frac{2\cdot5}{2\cdot6}=\frac{5}{6}[/tex]Is the prime factor of 121 11x11?
The prime factor of 121 is simply 11.
11x11 =121, since you can't take 11 two times.
The weight P in pounds that a beam can safely carry is inversely proportional to the distance D in feet between the supports of the beam. For a certain type of wooden beam,
K←
5,400
P= What distance between supports is needed to carry 900 lb?
D
uestion:
For the beam to support a weight of 900 lb, the distance between the supports needs to be no more than feet.
(Simplify your answer. Round to two decimal places as needed.)
The distance between supports is needed to carry 900 lb is 6 feet.
What is termed as the inversely proportional?Whenever the value with one quantity rises relative to the value of another, the two are referred to as being inversely proportional. It indicates that the two portions behave in nature in opposing ways. For instance, speed as well as time are inversely proportional to one another. The time decreases as the speed increases.For the given question,
The distance of the support of the beam is 'D'.
P ∝ 1/D
Now, weight is inversely proportional to distance.
P is the weight needed to carry = 900 lb.
Thus,
P = K/D
Where is proportionality constant; K = 5400.
D = K/P
Put the values.
D = 5400/900
D = 6 feet
Thus, the distance between supports is needed to carry 900 lb is 6 feet.
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Angel Corporation produces calculators selling for $25.99. Its unit cost is $18.95. Assuming a fixed cost of $80,960, what is the breakeven point in units?
The breakeven point is 11,500, meaning that if the sell that number of units, the profit will be zero.
How to get the breakeven point?
Here we know that the unit price is $25.99, so if they sell x units, the revenue is
R(x) = $25.99*x
And the cost per unit is $18.95, plus a fixed cost of $80,960
Then the cost of x units is:
C(x) = $80,960 + $18.95*x
The breakeven point is the value of x such that the cost is equal to the revenue, so we need to solve:
$25.99*x = $80,960 + $18.95*x
$25.99*x - $18.95*x = $80,960
$7.04*x = $80,960
x = $80,960/%7.04 = 11,500
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What is 120 plus 5% sales tax
Answer:
126
Step-by-step explanation:
Hello!
Since you are adding 5% of 120 to 120, we can simply find 105% of 120.
To do that, we can multiply 120 by the number value of the percentage.
Find the Total Price105% of 1201.05 * 120126The total price after the sales tax is 126.
Write the nth rule for the following geometric sequence. Then find the fifth term. (you are given the first term and the common ratio)1-
The formula for determining the nth term of a geometric sequence is expressed as
Tn = ar^(n - 1)
Where
a represents the first term
r represents the common ratio.
n represents the number of terms
From the information given,
a = 2, r = 3
Thus, the rule for the nth term of the geometric sequence is
Tn = 2 x 3^(n - 1)
To determine the fifth term, we would substitute n = 5 into the equation. It becomes
T5 = 2 x 3^(5 - 1)
T5 = 2 x 3^4
T5 = 162
The fifth term is 162
Use quadratic regression to find the equation of a quadratic function that fits the given points. x 0 1 2 3. Y. 49 50.4 39.5. 21
The regression Quadratic equation y = 49 + 7.55x - 6.15x².
What is Regression Equation?The technique of finding the equation of a parabola that most closely matches a collection of data is known as quadratic regression. The graph points that make up the parabola-shaped shape of this set of data are given. The parabola's equation is written as y = ax² + bx + c, where a never equals zero.
For data presented as ordered pairs, you can calculate the model's degree by identifying differences between dependent values. The model will be linear if the initial difference has the same value. The model will be quadratic if the second difference has the same value as the first.
As, we know the Quadratic model
Quadratic model, y = a + bx + cx²
Now, value of
a = 49
b = 7.55
c = -6.15
Then, the Regression Quadratic model is
y = 49 + 7.55x - 6.15x²
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The shorter leg of a right triangle is 9cm shorter than the longer leg. The hypotenuse is 9cm longer than the longer leg. Find the side lengths of the triangle.Length of the shorter leg: _ cmLength of the longer leg:__ cmLength of hypotenuse __ cm
Explanation:
let the longer leg = x
The shorter leg = 9cm shorter than the longer leg
The shorter leg = x - 9
hypotenuse = 9cm longer than the longer leg
hypotenuse = x + 9
Using pythagoras theorem:
hypotenuse² = shorter leg² + longer leg²
(x + 9)² = x² + (x - 9)²
Expanding:
x² + 9x + 9x + 81 = x² + x ² - 9x -9x + 81
x² + 18x + 81 = 2x² -18x + 81
collect like terms:
18x + 18x + 81 - 81 = 2x² - x²
36x + 0 = x²
x² - 36x = 0
x(x - 36) = 0
x = 0 or (x - 36) = 0
x = 0 or x = 36
if x = 0
shorter side = x - 9 = 0 - 9 = -9
Since the length cannot be negative, x = 36
The longer leg = x = 36 cm
The shorter leg = x - 9 = 36 - 9
The shorter leg = 27cm
The hypotenuse = x + 9 = 36 + 9
The hypotenuse = 45 cm
Which of the following shows a graph of a tangent function in the form y = atan(bx − c) + d, such that b equals one fourth question mark
ANSWER :
EXPLANATION :
PMark for Review 1 Harold spent the summer working at a diner. He now pays for a monthly subscription to a magazine. The equation y -35x + 180 can be used to represent this situation, where y is the amount of money Harold has remaining after x months of paying for his monthly magazine subscription. Which statement best describes the amount of money Harold has, given this equation? - A) Harold started with $35 and he spends $180 per month on his magazine subscription. B) Harold started with $180 and he gets paid $35 per month. C) Harold started with owed $180 for magazines and he continues to spends $35 per month on his magazine subscription. D) Harold started with $180 and he spends $35 per month on his magazine subscription.
the option D is the correct answer
the equation is
y = 35x +180
that means he started with 180 $ and his monthly subscription is 35 $.
Select the expressions that are equivalent to 7(7f)1. 49f2. 7(f+6f)3. f+144. f+49
ANSWER :
49f and 7(f + 6f)
EXPLANATION :
From the problem, we have :
[tex]7(7f)[/tex]When multiplied, it will be 49f
When breaking it down, 7f is equal to f + 6f. Then it will be 7(f + 6f)
The next options f + 14 and f + 49 has two terms, so it will not be equivalent to the given expression with one term.
So the only expressions that are equivalent to the given expression are 1 and 2
In a class of 36 students, 25
study Chemistry, 22 study
Maths and 25 study Physics, 17
study Physics and Maths,18
study Physics and Chemistry
and 15 study only one of the
three subjects. Find the;
a) number of students who
study all three subject?
b) number of students who
study only Maths and
Chemistry?
c) Probability that a student
selected at random study only
two of the three subjects?
a) Number of students who study all three subject = 15
b) Number of students who study only Maths and Chemistry = 1
c) Probability that a student selected at random study only two of the three subjects = 1/36
Define Probability
Simply put, probability is the likelihood that something will occur.
When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.Statistics is the study of events that follow a probability distribution.
As it is given total number of students is = 36
The subject are Physics, Maths, and Chemistry
Let, physics = p
maths = m
chemistry = c
The possible combination are,
p, c, m, pm, cp, cm, pcm (means 7 combination total)
Let x be the number of student who study all three subjects.
The students who study physics and maths = 17 - x
The students who study physics and chemistry = 18 - x
The number of student who study physics = 25
Now, with the expression we can find the students who study only physics
25 - ((x) + (18 -x) + (17- x))
⇒25 - (x + 18 - x + 17 - x)
⇒25 - (35 - x)
⇒25 - 35 + x
⇒x - 10
Let y be the number of student only chemistry and mathematics.
Now, with the expression we can find the students who study only chemistry
25-(x + (18- x)) + y
⇒25 - 18 + y
⇒ 7 - y
Now, with the expression we can find the students who study only maths
22 - (x + (17 - x)) + y
⇒ 22 - 17 + y
⇒ 5 - y
The possible combination and expression for each
pcm → x
cm → y
pc → 18 - x
pm → 17 - x
p → x - 10
c → 7 - y
m → 5 - y
____________
Total → 37 - y
But the number of students is 36 , so y = 1
That means,
The number of student who take only chemistry = 7 - y
= 7 - 1 = 6
The number of student who take only maths = 5 - y
= 5 - 1 = 4
The 15 students takes only one of the three subject
the number that take only physics is 5
so, x - 10 = 5
x = 15 (the student who takes all 3 subjects)
The student who takes only physics and chemistry = 18 - x
= 18 - 10 = 3
The student who takes only physics and Maths = 17 - x
= 17 - 15 = 2
To cross check put the values of x and y,
pcm → 15
cm → 1
pc → 3
pm → 2
p → 5
c → 6
m → 4
____________
Total → 36
Therefore, the answers :
a) Number of students who study all three subject = 15
b) Number of students who study only Maths and Chemistry = 1
c) Probability that a student selected at random study only two of the three subjects = ( 3 + 1 + 2) / 36 = 1/36
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A swim team consists of 5 boys and 5 girls. A relay team of 4 swimmers is chosen at random from the team members. What is the probability that 3 boys are selected for the relay team given that the first selection was a girl? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
In order to find the probability start the construction of the possible relay team, if the team is made by any 4 swimmers then
[tex]10\cdot9\cdot8\cdot7=5040[/tex]if the first member is a girl and the other three needs to be boys, the number of possibilities are
[tex]1\cdot5\cdot4\cdot3=60[/tex]then, divide in order to find the probability
[tex]\frac{60}{5040}=\frac{1}{84}[/tex]Rebecca must complete 15 hours of volunteer work. She does 3 hours each day.
For the linear equation that represents y, the hours Rebecca still has to work after x days, what does the y-intercept represent?
The y-intercept represents the hours Rebecca must work.
How to represent linear equation?Linear equation can be represented in slope intercept from, point slope form and standard form.
Therefore, in slope intercept form it can be represented as follows:
Hence,
y = mx + b
where
m = slopeb = y-interceptShe must complete 15 hours of volunteer work. She does 3 hours each day. Let's represent Rebecca situation in linear form.
where,
y = hours Rebecca still has to work
x = the number of days
Therefore,
y = 15 - 3x
The y-intercept is 15 which implies the number of hours she must complete.
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The number of hours Rebecca must work is represented by the y-intercept in the linear equation.
What is the linear equation?An equation is said to be linear if the power output of the variable is consistently one.
The linear equation is y = mx + c, where m denotes the slope and c is its intercept.
Given that she is required to put in 15 hours of volunteer work. Each day, she works three hours.
As per the given situation,
If x represents the number of days and y represents the number of hours she must work
So the linear representation shows Rebecca's situation will be:
y = 15 - 3x
Therefore, the number of hours Rebecca must work is represented by the y-intercept in the linear equation.
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What is the difference between chemical and physical change
Answer:
Step-by-step explanation:
In a physical change the appearance or form of the matter changes but the kind of matter in the substance does not. However in a chemical change, the kind of matter changes and at least one new substance with new properties is formed.
The graph of a toy car's speed y
over time x is a parabola that
shows a minimum speed of 2 m/s
after 3 seconds. After 5 seconds,
the car's speed is 3 m/s. What is
the equation in vertex form of the
parabola?
The equation in vertex form of the parabola is y=-1/30(x+23/2)²+529/120
Y axis represends the toy car's speed
X axis represents time
y=ax²+bx+c
c=0
y=ax²+bx
2=9a+3b multiplied with -5
-10 = -45a -15b........equation 1
3=25a+5b multiplied with 3
9 = 75a + 15b............equation 2
adding equation 1 and 2
9-10=75a-45a+15b-15b
30a=-1
a=-1/(30)
2=9×(-1/30)+3b
3b=2+3/10=23/10
b=23/30
y=-1/30 x²+23/30=-1/30(x²+23x+(23/2)²)+1/30 ×(529/4)
y=-1/30(x+23/2)²+529/120
Therefore, the equation in vertex form of the parabola is y=-1/30(x+23/2)²+529/120
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Find the 5th term of the arithmetic sequence -5x – 5, -123 – 8,- 19x – 11, ...Answer:Submit Answer
5x – 5, -123x – 8,
- 19x – 11, ...
Difference is =
lily ordered a set of green and brown pin.she received 35 pins, and 80% of them were green.How many green pins did lily receive?
In total there are 35 pins so that correspound to the 100%, so we can use a rule of 3 to solve it so:
[tex]\begin{gathered} 35\to100 \\ x\to80 \end{gathered}[/tex]So the equation is:
[tex]x=\frac{35\cdot80}{100}=28[/tex]So there are 28 green pins
find the volume and total surface area of a right circular cone whose base diameter is 10 cm and whose altitude is 20 cm.
SOLUTION
Given the question in the question tab, the following are the solution steps to calculate the required measurements.
Step 1: write the given parameters
[tex]\begin{gathered} \text{diameter}=10\operatorname{cm},\text{altitude}=\text{height}=20\operatorname{cm} \\ r=\frac{d}{2}=\frac{10}{2}=5\operatorname{cm} \end{gathered}[/tex]Step 2: Calculate the volume of the right circular cone
[tex]\begin{gathered} V=\frac{\pi r^2h}{3} \\ V=\frac{\pi\times5\times5\times20}{3} \\ V=\frac{500\pi}{3}=523.5987756 \\ V\approx523.5988\operatorname{cm}^3 \end{gathered}[/tex]Step 3: Calculate the total surface area of the right circular cone
[tex]\begin{gathered} \text{TSA}=\pi r(r+\sqrt[]{h^2+r^2)} \\ \text{TSA}=\pi(5)(5+\sqrt[]{20^2+5^2)} \\ \text{TSA}=5\pi(5+\sqrt[]{400+25)} \\ \text{TSA}=5\pi(5+\sqrt[]{425})=5\pi(5+20.61552813) \\ \text{TSA}=5\pi(25.615528134) \\ \text{TSA}=402.3677749 \\ \text{TSA}\approx402.3678cm^2 \end{gathered}[/tex]Hence, the volume and the total surface area of the given right circular cone are approximately 523.5988cm³ and 402.3678cm² respectively
Two towns are 1050 miles apart, a group of hikers start from each town and walk the trail toward each other. They meet after a total hiking time of 200 hours. If one group travels 1 1/2 miles Per hour faster than the other group, find the rate of each group
Answer:
Rate of the faster group = 3.38 miles per hour
Rate of the slower group = 1.88 miles per hour
Explanation:
Let x = rate of the slower group
Therefore the rate of the faster group will be x + 1 1/2 = x + 3/2 = x + 1.5
From the question, we're told that the two groups traveled for a total hiking time of 200 hours.
We know that distance = rate x time
So the distance of the slower group will be = 200x
And the distance of the faster group will be = 200(x + 1.5)
So if the distance between each town is 1050, we can then solve of x as shown below;
[tex]\begin{gathered} 200x+200(x+1.5)=1050 \\ 200x+200x+300=1050 \\ 400x=750 \\ x=\frac{750}{400} \\ x=1.88\text{ mph} \end{gathered}[/tex]Therefore the rate of the faster group = 1.88 + 1.5 = 3.38 mph.
Solve the equation. Check your solution.20x - 4= 50x + 2x=(Simplify your answer. Type an integer or a simplified fraction.)
20x - 4 = 50x + 2
Solve for x
20x - 50x =
3/3=_/21Fill the blank space with the answer
In the expression 3/3=_/21, it can be observed that 7 is multipled by denominator 3 in order to obtain 21 in in denominator. So same number, 7 is also multiplied with the numerator also.
[tex]\frac{3}{3}\times\frac{7}{7}=\frac{21}{21}[/tex]So, 21 is to be filled at blank space.
I need help with finding the rational approximation of 37 using perfect squares
SOLUTION
For rational approximation of 37, it means we are to obtain the close estimate for the square root of 37.
using perfect squares,
The perfect square number immediately lower than 37 is
[tex]36[/tex]The perfect square number immediately higher than 37 is
[tex]49[/tex]Then we set up the problem as in the image below
The distance between 36 to 37 is lower than the distance between 49 to 37, hence the rational aproximation of 37 will be closer to the square root of 36 than the square root of 49.
This accouunt for the sqaure root of 37 in the image above
[tex]\sqrt[]{37}=6.08\approx6.1[/tex]Therefore
The rational aprosimation of 37 using perfect square is 6.1
For the two numbers listed find two factors of the first number such that their product is the first number and there sum is the second number
Given:
Product of two numbers is 24
Sum of two numbers is -11
-8 and -3 are the two numbers.
Find the missing length of the triangle. 22 cm 17.6 cm The missing length is centimeters.
Write the equation for a line that is perpendicular to the given line and contain the following points. 12. X=-11Contains the point (-5, -7)equation:____
Purple line is perpendicular to given line (x = -11), and the equation for this lines is y = -7
find the coordinates of point P that lies on the line segment MQ, M(-9,-5) , Q(3,5), and partitions the segment at a ratio of 2 to 5
