Given the right triangle ABC with altitude BD drawn to the hypotenuse AC. If AC=6 and DC=4, what is the length of BC in simplest radical form ?
Answers
This problem is an application of the Geometric mean theorem. It says that
[tex]\frac{6}{x}=\frac{x}{4}[/tex]Comment: In other words, it says that the length of BC (x) is the geometric mean between the lengths of AC and DC.
Then,
[tex]x^2=6\cdot4=24[/tex][tex]x=\sqrt[]{24}=2\cdot\sqrt[]{6}[/tex]................................................................................................................................................................
Let's talk a little about the simplest radical form of a square root
[tex]\sqrt[]{a}[/tex]The first step to finding it is to write the number within the root as a product of prime powers, such product is called its integer factorization. Let's do that for 24:
Then, the integer factorization of 24 is
[tex]24=2^3\cdot3[/tex]Thus,
[tex]\sqrt[]{24}=\sqrt[]{2^3\cdot3}[/tex]The idea now is to take out of the root all we can. The rule is that we can only take out powers of 2 (for our root is a square root). In the expression
[tex]2^3\cdot3[/tex]There is only one power of 2, within 2^3. We can write it as
[tex]2^2\cdot2\cdot3[/tex]How are we going to take out it? We are going to take out the base of the power, which is 2 in this case. Then,
[tex]\sqrt[]{24}=2\cdot\sqrt[]{2\cdot3}=2\cdot\sqrt[]{6}[/tex]In simple terms, the simplest radical form of a root is what results after taking out the root all that can be taken out.
Related Questions
WILL MARK BRAINLIEST. Two functions are given below: f(x) and h(x). State the axis of symmetry for each function and explain how to find it
Answers
Answer:
The axis of symmetry is -4 because the axis of symmetry is equal to h, in the vertex form!
20 pts, precalc, see attach
If f (x) = 3x2 + 5x − 4, then the quantity f of the quantity x plus h end quantity minus f of x end quantity all over h is equal to which of the following?
Answers
The numeric value for the given expression is as follows:
[tex]\frac{j(x + h) - j(x)}{h} = \frac{4^{x - 2}(4^h - 1)}{h}[/tex]
How to find the numeric value of a function or of an expression?To find the numeric value of a function or of an expression, we replace each instance of the variable in the function by the desired value.
In the context of this problem, the function j(x) is given as follows:
[tex]j(x) = 4^{x - 2}[/tex]
At x = x + h, the numeric value of the function is found replacing the lone instance of x by x + h as follows:
[tex]j(x + h) = 4^{x + h - 2}[/tex]
For the fraction, the subtraction at the numerator is given as follows, applying properties of exponents:
[tex]j(x + h) - j(x) = 4^{x + h - 2} - 4^{x - 2} = 4^{x - 2}(4^h - 1)[/tex]
As the x - 2 term is common to both exponents.
Just dividing by h, the numeric value of the entire expression is given as follows:
[tex]\frac{j(x + h) - j(x)}{h} = \frac{4^{x - 2}(4^h - 1)}{h}[/tex]
Which means that the third option is correct.
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twice a number increased by twenty is at least eighty-five. select all possible value of the number.
31
35
32
30
33
40
20
34
Answers
The number when increased by twenty is at least 85, the possible values of the numbers for this are: 35, 33, 40 and 34.
Given, according to the statement in the question, frame the equation:
2x+20 ≥ 85
⇒ 2x + 20 ≥ 85
⇒ 2x ≥ 85 - 20
⇒ 2x ≥ 65
⇒ x ≥ 65/2
⇒ x ≥ 32.5
hence the numbers greater than or equal to 32.5 are 35, 33, 40 and 34.
Hence the possible values of the number are 35, 33, 40 and 34.
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6+4×2+11+10÷2 order
Answers
To solve the following calculation
[tex]6+4\cdot2+11+10\div2[/tex]You have to keep in mind the order of operations. Which states that multiplications and divisions are to be solved before additions or subtractions.
So the first step is to solve the multiplication "4*2" and the division "10÷2"
[tex]6+8+11+5[/tex]Now all that is left is to add the numbers when you have to perform the same operation several times, you have to solve them from lef
— 3х + 2 = -4х + 4 need help
Answers
Given the following equation:
[tex]-3x+2=-4x+4[/tex]You need to solve for "x" in order to find its value and solve the equation. To do this, you can follow the steps shown below:
1. You need to apply the Subtraction property of equality by subtracting 2 from both sides of the equation:
[tex]\begin{gathered} -3x+2-(2)=-4x+4-(2) \\ -3x=-4x+2 \end{gathered}[/tex]2. Now you can apply the Additio property of equality by adding "4x" to both sides of the equation:
[tex]\begin{gathered} -3x+(4x)=-4x+2+(4x) \\ x=2 \end{gathered}[/tex]Therefore, you get that the solution is:
[tex]x=2[/tex]There are two box containing only yellow and black pens
Answers
SOLUTION; Concept
Step1: Identify the giving information in the question
BOX A contains
[tex]\begin{gathered} 9\text{ yellow pens} \\ 6\text{ Black pens } \end{gathered}[/tex]BOX B contains
[tex]\begin{gathered} 9\text{ yellow pens } \\ 11\text{ black pens} \end{gathered}[/tex]Step2: Find the probability of each event
Event 1: Choosing a green pen from the Box B
[tex]\begin{gathered} \text{ Since there is no gr}een\text{ pen in the box, then probability of choosing a gr}en\text{ box in Box B is 0} \\ \text{then probability of choosing a gr}en\text{ box in Box B is 0} \\ Pr(E1)=0 \end{gathered}[/tex]Event 2: Choosing a black pen from the Box B
[tex]P(E2)=\frac{11}{9+11}=\frac{11}{20}=0.55[/tex]Event 3: Choosing a yellow or black pen from the Box A
Since Box A contains only a yellow or black pen then the probability is
[tex]Pr(E3)=1[/tex]Event 4: Choosing a yellow pen from box A
Since there are 9 yellow pens in box A, the probability of choosing the yellow pen is
[tex]Pr(E4)=\frac{9}{9+6}=\frac{9}{15}=0.6[/tex]Probability describes the likelihood of the event.
Hence From least likely to most likely the occurrence of the event is arranged as follows according to the probability of each event
[tex]\text{Event }1\rightarrow\text{ Event 2}\rightarrow\text{ Event 4}\rightarrow\text{ Event 3}[/tex]
A school offers band and chorus classes. The table shows the percents of the 1200 students in the school who are enrolled in band, chorus, or neither class. How many students are enrolled in both classes?
Class Enrollment
Band 34%
Chorus 28%
Neither 42%
Answers
168 students are enrolled in both classes.
This is a problem from set theory. We can solve this problem by following a few steps easily.
First of all, we have to calculate the students present in both classes.
Student present in both classes = the total student - the student not enrolled in both classes.
So the percentage of the students enrolled in both classes or any of one class is ( 100% - 42% ) = 58%.
Now, the students only enrolled in chorus class is ( 58% - 34%) = 24%
So, the students who joined both classes is ( 28%- 24%)= 4%
The total student is 1200, then 4% of the total student is
( 1200 × 14 )/100 = 168 students.
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Find the average rate of change of f(x) = - 2x ^ 2 - x from x = 1 to x = 6 . Simplify your answer as much as possible .
Answers
The average rate of change of a function in the interval [a,b] is given by:
[tex]\frac{f(b)-f(a)}{b-a}[/tex]In this case we have that a=1 and b=6; plugging these values in the formula above we have:
[tex]\begin{gathered} \frac{-2(6)^2-6-(-2(1)^2-1)}{6-1}=\frac{-2(36)-6-(-2(1)-1)}{5} \\ =\frac{-72-6-(-2-1)}{5} \\ =\frac{-78-(-3)}{5} \\ =\frac{-78+3}{5} \\ =\frac{-75}{5} \\ =-15 \end{gathered}[/tex]Therefore, the average rate of change in the interval is -15
2) The Allen's rectangular backyard has a
perimeter of 144 feet. If the backyard is 40
feet wide, what is the area of their yard?
Answers
Answer:
1280 ft²
Step-by-step explanation:
Perimeter is the addition of all sides of the shape added together. Therefore, if the backyard is the rectangular shape you know that two of the sides are 40 feet wide. By adding both sides you get a total of 80 feet. Subtract the total from the perimeter in order to get the total of the unknown sides. 144 minus 80 is equivalent to 64. Now that you have the total of the unknown sides, divide by two in order to get the single unknown length. So, 64 divided by 2 is equal to 32. The area is the multiplication of two sides with quadrilaterals. Therefore, 40 times 32 equals an area of 1280 feet squared.
Translate the following sentence into an equation.
Twelve minus three times x equals fourteen.
Answers
Answer:
12-3x=14
Step-by-step explanation:
Rewrite the decimal fraction as a decimal number.
14 25
100
Answers
The decimal number of the fraction [tex]14\frac{25}{100}[/tex] is 14.25
The given mixed fraction = [tex]14\frac{25}{100}[/tex]
The mixed fraction is the fraction that consist of one whole number and the simple fraction.
The simple fraction is the fraction that consist of numerator and denominator as whole number. The top term of the simple fraction is called numerator and bottom term is called denominator
The decimal number is a number that consist of one whole number and the fractional part. The fractional parts are separated by a decimal point.
The number is [tex]14\frac{25}{100}[/tex]
Convert the mixed fraction to simple fraction
[tex]14\frac{25}{100}[/tex] = (100×14+25)/100
= 57/4
Convert the simple fraction to decimal number
57/4 = 14.25
Hence, the decimal number of the fraction [tex]14\frac{25}{100}[/tex] is 14.25
The complete question is:
Rewrite the decimal fraction as a decimal number.
[tex]14\frac{25}{100}[/tex]
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1,857.205 round each number to the place of the underlined digit.0 is the underlined digit
Answers
Answer:
1,857.21
Explanation:
In the number: 1,857.205
The digit after 0 is 5.
Since it is a number between 5 and 9, we round up to obtain:
[tex]1,857.205\approx1,857.21\text{ (to the nearest hundredth)}[/tex]
Juan took out a $5000 loan for 292 days and was charged simple interest.
The total interest he paid on the loan was $336 As a percentage, what was the annual interest rate of Juan's loan?
Assume that there are 365 days in a year
Answers
Since the total interest he paid on the loan was $336. Then, The annual interest rate of Juan's loan if there are 365 days in a year will be as at 8.4%.
What is Simple Interest?Simple Interest (S.I.) is the method of calculating the interest amount for a particular principal amount of money at some rate of interest.
For example, when a person takes a loan of Rs. 5000, at a rate of 10 p.a. for two years, the person's interest for two years will be S.I. on the borrowed money.
The formula of simple interest is given as:
S.I = PRT
Where:
P = principalR = rateT = timeSubstituting the values and solving for the rate on the loan
r = {A} {PT}\r
= {5336} {5000*292}
r = 8.4 %
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Dog Owners5. A city council wants to know if residents would like a dogpark. They sent a survey to every household in the city.The results of those who responded are shown in the tableat the rightNumbexof Dogs inHouseholdNumber ofHouseholds0513a What is an appropriate first step in finding the experimentalprobability that a household has 2 or more dogs?12182 or more129Find the product of the number of households withone dog and the number with two or more dogs.Find the difference of the number of households with twoor more dogs and the number with no dogs.© Find the sum of the number of households for each category.Find the difference of the number of households with no dogsand the number with one dog or more.b. What is the experimental probability that a householdhas 2 or more dogs?
Answers
Given:
A table represents a survey to know if residents would like a dog.
a) What is an appropriate first step in finding the experimental probability that a household has 2 or more dogs?
The first step is to find the total households
So, the answer will be option C
Find the sum of the number of households for each category.
b) What is the experimental probability that a household has 2 or more dogs?
First, the total number of households = 513 + 218 + 129 = 860
And the number of households has 2 or more dogs = 129
So, the probability = 129/860 = 0.15 = 15%
So, the answer will be 15%
Where X in the age of the baby in months according to this model what is the weight in pounds of a baby At age 5 months
Answers
The given function is:
f(x) = 1.5x + 7
Then, since x represents the age of the baby, in months, in order to find its weight with 5 months, that is, f(5), we need to replace x by 5 in the above equation:
f(5) = 1.5 * 5 + 7 = 7.5 + 7 = 14.5
Therefore, the last option is correct.
What is the P(A and B) if P(A) = 1/2 and P(B) = 2/7, where A and B are independent events?5/81/71/121/2
Answers
EXPLANATION:
To calculate the number of independent events that occur, the product of the probabilities of the individual events occurring must be calculated.
Therefore if A and B are independent events then:
P (A and B) = P (A) • P (B)
The exercise is as follows:
[tex]\begin{gathered} \frac{1}{2}\times\frac{2}{7}=\frac{2}{14};\text{ Now }we\text{ must take }square\text{ r}oot; \\ \frac{2}{14}=\frac{1}{7} \\ \text{ANSWER: }\frac{1}{7} \end{gathered}[/tex]NOTE:
To obtain the product of two fractions, the numerators must be multiplied with each other and the denominators must also be multiplied with each other.
On average, Peter goes through three fish hooks in order to catch 7 fish. How many hooks can he expect to use if he needs to catch 189 fish?
Answers
By solving a proportional relation, we conclude that he needs 81 hooks to catch 189 fish.
How many hooks can he expect to use if he needs to catch 189 fish?We assume there is a proportional relationship of the form:
F = k*H
where:
F = number of fish.k = constant of proportionality.H = number of hooks.We know that with 3 hooks he catches 7 fish, then we can replace that:
7= k*3
7/3 = k
So the proportional relation is:
y = (7/3)*x
Then if he wants to get 189 fish we can write:
189 = (7/3)*x
And solve this for x:
189*(3/7) = x =81
He will need 81 hooks.
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An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 245ft. Use the formula s=24d−−−√ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.
Answers
The vehicle speed is 53 units if s = 24d and d = 245ft
Given the formula for speed is = √24 d
s denotes the vehicle's speed.
d = the length of the skid marks
We need to find the speed of the vehicle.
We know that the displacement d is 117 ft
d = 117 ft
And also it is mentioned in the question to use the below-given formula.
s = √24d
Now substituting the value of displacement in the formula we get
=> s = √24 x 117
=> s = √2808
=> s = 52.99
Now approximating to the nearest value of the decimal, we get.
=> s = 53
Therefore the speed of the vehicle is 53 units.
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Of the 100 million acres in California,
the federal government owns 45 million
acres. What percent is this?
Answers
Step-by-step explanation:
you do know what a percent is ?
1% is the 1/100 part of a whole.
when we have
100,000,000 (one hundred million),
how many 1/100 parts of that are
45,000,000 (45 million)?
well, 45.
45,000,000 is 45/100 of 100,000,000
in other words 45%.
the variable y varies directly as x. when x =20, y= 12 what is the value of y when x = 15a:7b:9c:18d:25
Answers
To direct variations use a rule of three, as follow:
[tex]\begin{gathered} \frac{?}{12}=\frac{15}{20} \\ \\ ?=12*\frac{15}{20} \\ \\ ?=\frac{180}{20} \\ \\ ?=9 \end{gathered}[/tex]Then, when x=15, y is 9Answer: B.9The height, h (in meters above ground), of a projectile at any time, t (in seconds), after the launch is defined by the function h(t) = −5t2 + 13t + 6. The graph of this function is shown below. When rounded to the nearest tenth, what is the maximum height reached by the projectile, how long did it take to reach its maximum height, and when did the projectile hit the ground?
Answers
As we can se from the graph, the maximum value of the function is almost 15 meters, it is reached between t=1s and t=2s, and the height is 0 again when the time is approximately 3 seconds.
We can find the exact values by using the equation.
First, find the zeroes of the function by setting h(t)=0 and solving the quadratic equation for t:
[tex]\begin{gathered} h(t)=-5t^2+13t+6=0 \\ \Rightarrow t=\frac{-13\pm\sqrt[]{13^2-4(6)(-5)}}{2(-5)} \\ \Rightarrow t_1=-0.4;t_2=3 \end{gathered}[/tex]Then, the projectile hits the ground at t=3s.
To find the time at which the projectile reaches its maximum height, find the average between both zeroes:
[tex]t_{\max }=\frac{t_1+t_2}{2}=\frac{-0.4s+3s}{2}=1.3s[/tex]To find the maximum height, evaluate h at t=1.3s:
[tex]h(1.3s)=-5(1.3)^2+13(1.3)+6=14.45[/tex]Therefore, to the nearest tenth, the projectile reached a maximum height of 14.5 meters in 1.3 seconds and it took 3.0 seconds for it to hit the ground.
The correct option is the second one.
Write the equation of the line, with the given properties, in slope intercept form. Slope = -7, through (-6,8)
Answers
Given:
The slopr of line is m = -7.
The line passes through point (-6,8).
Explanation:
The equation of line in slope-intercept form is,
[tex]y=mx+c[/tex]Substitute the valus in the equation to determine the value of c.
[tex]\begin{gathered} 8=-7\cdot(-6)+c \\ c=8-42 \\ =-34 \end{gathered}[/tex]So equation of line is y = -7x - 34.
Find a solution for the equation. x^2-6x-8=0
Answers
Given:
[tex]x^2-6x-8=0[/tex]Required:
To find the solution of the given equation.
Explanation:
From the given equation,
[tex]\begin{gathered} a=1 \\ b=-6 \\ c=-8 \end{gathered}[/tex]Now
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]Therefore,
[tex]x=\frac{-(-6)\pm\sqrt{(-6)^2-4(1)(-8)}}{2(1)}[/tex][tex]\begin{gathered} x=\frac{6\pm\sqrt{36+32}}{2} \\ \\ =\frac{6\pm\sqrt{68}}{2} \\ \\ =\frac{6\pm2\sqrt{17}}{2} \\ \\ =\frac{2(3\pm\sqrt{17)}}{2} \\ \\ =3\pm\sqrt{17} \end{gathered}[/tex]Final Answer:
The solutions are
[tex]x=3+\sqrt{17},3-\sqrt{17}[/tex]Write the point slope form of the line satisfying the given conditions. Then use the point slope form of the equation to write the slope intercept form of the equation. Slope=7Passing through (-6,1)
Answers
Ok, so
The point slope form of the line is given by the following formula:
[tex](y-y_1)=m(x-x_1)[/tex]Where
[tex](x_1,y_1)[/tex]Is a point of the line, and m is the slope.
If we replace our values:
Slope = 7
Point = (-6, 1)
We obtain that the equation is:
[tex]\begin{gathered} (y-1)=7(x-(-6)) \\ (y-1)=7(x+6) \end{gathered}[/tex]To find the slope intercept form of the equation, we distribute in the brackets:
[tex]\begin{gathered} y-1=7x+42 \\ y=7x+43 \end{gathered}[/tex]And the equation of our line in the slope intercept form will be:
y=7x+43
What is the square root of 225As I am new to thisPlease answer step by step in the easiest form possible
Answers
Given:
[tex]225[/tex]To Determine: The square root of the given number
Solution
Step 1: Express 225 as the product pf its prime factors of 225
[tex]\begin{gathered} 225=3\times3\times5\times5 \\ 225=3^2\times5^2 \end{gathered}[/tex]Step 2: Separate the factors into their own square root
[tex]\begin{gathered} \sqrt{225}=\sqrt{3^2\times5^2} \\ \sqrt{225}=\sqrt{3^2}\times\sqrt{5^2} \end{gathered}[/tex]Step 3: Solve the individual's squares
[tex]\begin{gathered} \sqrt{225}=\sqrt{3^2}\times\sqrt{5^2} \\ \sqrt{225}=3\times5=15 \end{gathered}[/tex]Hence, the square root of 225 is 15
You will purchase snacks for the painting class. You have abudget of $40. You want to buy fruit and granola bars. Fruit costs $4 perpound, and the granola bars are $1 each. You need at least 20 granolabars. What combinations of fruit and granola bars can you buy?My variables- f-fruits g-granola
Answers
Budget: $40
Fruit: $4/pound
Granola bars: $1 each
At least 20 granolas
Let's call
f: number of pounds of fruit
g: number of granola bars
Therefore
The total cost of fruit is given by: $4 · f
The total cost of granola is given by: $1 · g
The total cost of everything is given by: $4 · f + $1 · g
Since budget: $40 then
$4 · f + $1 · g = $40
4f + g = 40
g > 20 means we have at least 20 granolas
4f + g > 20 means that we have at least 20 granolas, that cost $20, since 4f + g = 40 and 40 > 20
Find the minimum value of
C = 6x + 3y
Subject to the following constraints:
x > 1
y ≥ 1
4x + 2y < 32
2x + 8y < 56
Answers
Answer:
9
Step-by-step explanation:
You want the minimum value of objective function C=6x+3y, given the constraints x>1, y≥1, 4x+2y<32, and 2x+8y<56.
MinimumThe objective function has positive coefficients for both x and y, so it will be minimized when x and y are at their minimum values. The constraints tell you these minimum values are x=1 and y=1, so the minimum value of C is ...
C = 6(1) +3(1) = 9
The minimum value of C is 9.
__
Additional comment
The value of x cannot actually be 1, so the value of C cannot actually be 9. However x may be arbitrarily close to 1, so C may be arbitrarily close to 9.
C = 6x +3y ⇒ x = (C -3y)/6
The x-constraint requires ...
x > 1
(C -3y)/6 > 1
C -3y > 6 . . . . . . multiply by 6
C > 6 +3y . . . . . . add 3y
The minimum value of y is exactly 1, so we have ...
C > 6 +3(1)
C > 9
What bearing and airspeed are required for a plane to fly 360 miles due north in 2.0 hours if the wind is blowing from a direction of 331 degrees at 15 mph?
Answers
Using the vector form of velocity, it is possible to calculate the airspeed and bearing, which are 165.33 mph and 0.7 degrees respectively.
Given:
A plane may go 360 miles straight north in 2.0 hours if the wind is blowing at 15 mph and 331 degrees from the north.
You can use the following formula to calculate airspeed.
[tex]V^{2} =V_{x} ^{2} + V_{y} ^{2}- 2v_{x} v_{y}cos\beta \\[/tex] ...........equation (1)
[tex]v_{x}[/tex] is the velocity of the wind and [tex]v_{y}[/tex] is the velocity of the plane.
[tex]\beta = 331-180 = 151 degree[/tex]
Putting [tex]v_{x}, v_{y} , \beta[/tex] in equation (1)
[tex]V^{2}= 15^{2} +180^{2} -2*15*180*cos151\\ V^{2} = 225 + 32400 - 5400*cos151\\V^{2} = 27336.49\\V = 165.33[/tex]
Hence, The airspeed of the plane is 165.33 mph
Now,
[tex]\frac{sin\alpha }{v_{x} } =\frac{sin\beta }{V} \\\frac{sin\alpha }{15 } =\frac{sin151 }{165.33} \\\\sin\alpha = 10*0.00122\\\alpha = 0.7[/tex]
Hence bearing required is 0.7 degrees.
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please slove this for me-3(x+1)<15
Answers
-3(x+1)<15
Divide both-side of the inequality by -3
(x + 1) > -5
subtract 1 from both-side of the inequality
x > -5 - 1
x > -6
Answer:
Step-by-step explanation:
-3(x+1)<15
Divide the inequality by -3, so we need to change the sides of the less than into greater than
x+1 > -5
x > -5 - 1
x > -6
64x power 9 as a cube of a monomial
Answers
Answer:21
Step-by-step explanation: