Answer:
[tex]\large\boxed{234\,\text{students}}[/tex]
Step-by-step explanation:
In this question, we're trying to find how many students are going to the homecoming dance.
To find the answer, we need to use some information that was provided to us from the question.
Important information:
72% of high school students are attending the homecoming danceThere are 325 students at the schoolWith the information above, we can solve the question.
To make this simple, we're going to need to figure out how much of 325 is 72%, due to the fact that we need to find the 72% of students that are going to the dance (out of 325 students).
To do this, we would multiply 325 by 0.72
[tex]325*0.72=234[/tex]
When you multiply, you should get 234.
This means that 234 students from the school are going to the dance.
I hope this helped you out.Good luck on your academics.Have a fantastic day!Approximately 234 out of 325 students at Wilson High School, which constitutes about 72% of the whole body, are attending the homecoming dance.
Explanation:The subject of this question is mathematics, specifically a practical application of percentage calculations. In this scenario, we need to determine the number of students from Wilson High School attending the homecoming dance if 72% of the total student body, which comprises 325 students, is attending.
To solve this, we multiply the total number of students (325) by the percentage of students attending the dance in decimal form (0.72). So, 325 * 0.72 will give us the number of students attending the dance.
After calculating, we find that 234 (rounded to the nearest whole number) students are planning to attend the homecoming dance at Wilson High School.
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URGENT!!!!!Driving times for students' commute to school is normally distributed, with a mean time of 14 minutes and a standard deviation of 3 minutes. Using the empirical rule, approximately what percent of students' commute time is between 11 and 17 minutes? 32% 68% 95% 99.7%
Answer:
B. 68%.
Step-by-step explanation:
We have been given that driving times for students' commute to school is normally distributed, with a mean time of 14 minutes and a standard deviation of 3 minutes.
First of all, we will find z-score of 11 and 17 using z-score formula.
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{11-14}{3}[/tex]
[tex]z=\frac{-3}{3}[/tex]
[tex]z=-1[/tex]
[tex]z=\frac{17-14}{3}[/tex]
[tex]z=\frac{3}{3}[/tex]
[tex]z=1[/tex]
We know that z-score tells us a data point is how many standard deviations above or below mean.
Our z-score -1 and 1 represent that 11 and 17 lie within one standard deviation of the mean.
By empirical rule 68% data lies with in one standard deviation of the mean, therefore, option B is the correct choice.
Answer: 68%
Step-by-step explanation: ya boy just took le test :-)
Could someone help me with this math problem?
Answer:
729
Step-by-step explanation:
1/3^-2×3^-4×(-1)^2
=3^2×3^4/1
=9×81
=729
For this case we have the following expression:
[tex]\frac {1} {3^ {- 2} * x^{ - 4} * y ^ 2}[/tex]
We must evaluate the expression to:
[tex]x = 3\\y = -1[/tex]
So:
[tex]\frac {1} {3^{- 2} * 3^{ - 4} * (- 1) ^ 2} =[/tex]
[tex]\frac {1} {\frac {1} {3 ^ 2} * \frac {1} {3 ^ 4} * 1} =\\\frac {1} {\frac {1} {3 ^ 2} * \frac {1} {3 ^ 4}} =\\\frac {1} {\frac {1} {9} * \frac {1} {81}} =\\\frac {1} {\frac {1} {729}} =\\\frac {729} {1} =\\729[/tex]
Answer:
Option B
given the function f(x) =2x-5 and g(x) which function has a greater slope
x g(x)
2 0
4 5
6 10
A. f(x) has a greater slope
B. g(x) has a greater slope
C. the slopes of f(x) and g(x) are the same
D. the slope of g(x) is undefined
Answer:
B. g(x) has a greater slope.Step-by-step explanation:
The slope-intercept of an equation of a line:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
=========================================
[tex]f(x)=2x-5\to m=2[/tex]
From the table of function g(x) we have:
x = 2 → y = 0
x = 4 → y = 5
Calculate the slope:
[tex]m=\dfrac{5-0}{4-2}=\dfrac{5}{2}=2.5[/tex]
The slope of f(x) is equal to 2.
The slope of g(x) is equal to 2.5.
2 < 2.5
Answer:
B. g(x) has a greater slope
Step-by-step explanation:
Given the function f(x) =2x-5 and g(x), g(x) has a greater slope.
f(x) = 2
g(x) = 2.5
Which best describes the solutions to the inequality x>10?
(A)10 and every whole number greater than 10
(B)a rational number infinitely close to 10 but greater than 10, and all other rational numbers greater than 10
(C)11 and every whole number greater than 11
(D)a rational number infinitely close to 10 but greater than 10, and all other whole numbers greater than 10
Answer:
It can be B or C (B is closer to being correct); neither are worded perfectly correct. Definitely not A or C.
Step-by-step explanation:
I am a math teacher and whoever created this question didn't cover everything.
The answer is all numbers greater than 10, not just rational (irrational should also be included) and not just whole numbers (fractions and decimals should be included).
Answer:
(B)a rational number infinitely close to 10 but greater than 10, and all other rational numbers greater than 10
Step-by-step explanation:
x> 10 means all numbers greater than 10
(A)10 and every whole number greater than 10
False, does not include 10
(B)a rational number infinitely close to 10 but greater than 10, and all other rational numbers greater than 10
True
(C)11 and every whole number greater than 11
False, it only included integers. 10.5 is a solution but not included here
(D)a rational number infinitely close to 10 but greater than 10, and all other whole numbers greater than 10
False, it only included whole numbers. 10.5 is a solution but not included here
The answer should really be real numbers, not rational numbers.
Irrational numbers can be solutions. But given the choices given, B is the best solution.
what is the simplified form of sqaure root 72x to the power 16 over 50x 36 assume x = 0
1)6 over 5x power of 10
2)6 over 5x to power of 2
3)6 over 5x to the power of 10
4)6 over 5x to the power of 2
Answer:
[tex]\large\boxed{\dfrac{6}{5x^{10}}}[/tex]
Step-by-step explanation:
[tex]\sqrt{\dfrac{72x^{16}}{50x^{36}}}\qquad\text{simplify}\\\\=\sqrt{\dfrac{36x^{16}}{25x{^{20+16}}}}\qquad\text{use}\ (a^n)(a^m)=a^{n+m}\\\\=\sqrt{\dfrac{36x^{16}}{25x^{20}x^{16}}}\qquad\text{cancel}\ x^{16}\\\\=\sqrt{\dfrac{36}{25x^{20}}}\qquad\text{use}\ \sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}\ \text{and}\ \sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\\\\=\dfrac{\sqrt{36}}{\sqrt{25}\cdot\sqrt{x^{10\cdot2}}}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=\dfrac{6}{5\sqrt{(x^{10})^2}}\qquad\text{use}\ \sqrt{a^2}=a\ \text{for}\ a\geq0\\\\=\dfrac{6}{5x^{10}}[/tex]
the sum of to numbers is 15 and their quotient is 2.
PLEASE HELP MEEE
Answer:
The two numbers are 5 and 10.
Step-by-step explanation:
We need to solve the following system of equations:
We know that:
A + B = 15
A/B = 2
Solving the system of equations we have:
A/B = 2 ⇒ A = 2B
Then:
2B + B = 15 ⇒ 3B = 15 ⇒ B = 5
Then, we need to find A:
A = 10
Answer:
The two numbers are 5 and 10
Step-by-step explanation:
Let x be one number and y be the other number
Their sum is 15
x+y = 15
The quotient is 2
x/y =2
Rewriting this equation by multiplying by y
x/y * y = 2*y
x = 2y
Substitute this into the first equation
2y+ y = 15
Combine like terms
3y = 15
Divide by 3 on each side
3y/3 =15/3
y=5
Now we can find x
x = 2y
x =2(5)
x=10
Help with number two
Answer:
y = -2x+1
Step-by-step explanation:
We have a point and a slope, so we can use the point slope form of a line
y-y1 = m(x-x1) where m is the slope and (x1,y1) is the point
y-3 = -2(x--1)
y-3=-2(x+1)
Distribute the 2
y-3 = -2x-2
Add 3 to each side
y-3+3 = -2x-2+3
y = -2x+1
This is in slope intercept form
You need a 45% alcohol solution. On hand, you have a 350 mL of a 15% alcohol mixture. You also have 70% alcohol mixture. How much of the 70% mixture will you need to add to obtain the desired solution?
You will need
_____ mL of the 70% solution
Answer:
420
Step-by-step explanation:
Amount in the 45% solution + amount in the 70% solution = amount in the 45% solution
0.15 × 350 + 0.70 × V = 0.45 × (350 + V)
52.5 + 0.70V = 157.5 + 0.45V
0.25V = 105
V = 420
You need 420 mL of the 70% solution.
After setting up and solving a weighted average equation, the result reveals that you would need 1050 mL of the 70% alcohol solution to achieve the desired 45% alcohol solution.
Explanation:To solve this problem, we can use the concept of weighted averages. The final volume of alcohol in the 45% solution will be the sum of the alcohol in the 15% solution and the 70% solution. Let's denote the volume of the 70% solution we need to add as X ml. So, the equation will be:
0.15 × 350 + 0.70 × X = 0.45 × (350 + X)
Solving this equation will give us the amount of 70% solution needed. After simplifying, you get:
52.5 + 0.7X = 157.5 + 0.45X
Further simplification gives:
0.7X - 0.45X = 157.5 - 52.5
Therefore, X = 1050 ml. So, you will need 1050 ml of the 70% solution.
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Henry, Brian and Colin share some sweets in the ratio 5:4:2. Henry gets 15 more sweets than Colin. How many sweets does Brian get?
Answer:
20 sweets.
Step-by-step-explanation:
Let Colin have x sweets.
The Henry gets x+15 sweets
Then according to the ratios:
5/2 = x+15/x
5x = 2x + 30
3x = 30
x = 10.
So Colin has 10 sweets.
The ratio of Brian's sweets to Colin's sweets is 4: 2 or 2:1.
So Brian has 2 * 10 = 20 sweets.
The cat’s weight changed -8 oz. while she was sick. Which of the following shows a greater change in weight? A. Loss of 9 oz B. Loss of 6 oz. C. Gain of 5 oz. D. Gain of 3 oz.
Answer:
The correct answer option is A. Loss of 9 oz.
Step-by-step explanation:
We are given that a cat's weight change -8 oz. while she was sick. It means that the cat lost 8 ounces of weight.
We are to determine whether which of the given answer options show a greater change in weight.
The correct answer for this is: loss of 9 oz which is a greater loss than 8 oz.
8c - c +6=48. How do I explain this with words?
PLEASE HELP URGENT!!!! what is the measure of angle C? 38 degrees. 76 degrees. 90 degrees. 152 degrees.
Answer:
38
Step-by-step explanation:
less than 45
Answer:38
Step-by-step explanation:
0.7 of 12.99
how do you solve it?
Answer:
9.093
Step-by-step explanation:
Of means multiply
.7 * 12.99
9.093
Answer:
9,093
Step-by-step explanation:
Yes. You take 70% of 12,99 [multiply].
I am joyous to assist you anytime.
In 1995, the moose population in a park was measured to be 4200. By 1998, the population was measured again to be 1600. If the population continues to change linearly:
Find a formula for the moose population, P, in terms of t, the years since 1990.
P=
What does your model predict the moose population to be in 2003?
Answer:
P = -2600/3 t + 25600/3
P = -8200/3
Step-by-step explanation:
t is the time in years since 1990, so two points on the line are (5, 4200) and (8, 1600).
Using the points to find the slope:
m = (y₂ − y₁) / (x₂ − x₁)
m = (1600 − 4200) / (8 − 5)
m = -2600/3
Now writing the equation in point-slope form:
P − 4200 = -2600/3 (t − 5)
Converting to slope-intercept form:
P − 4200 = -2600/3 t + 13000/3
P = -2600/3 t + 25600/3
In 2003, t = 13:
P = -2600/3 (13) + 25600/3
P = -8200/3
The linear formula for the moose population is P = -800t + 8200. The moose population predicted by this model for the year 2003 is 2,400.
Explanation:In this question, we are given that in 1995 the moose population was 4200 and by 1998 it was 1600. This change in population mimics a linear relationship. We are asked to find the formula for this line and then predict the moose population in 2003.
We know that 1995 corresponds to t = 5 (since t is the years since 1990) and 1998 corresponds to t = 8. Therefore we can find the slope of the line (m) as (4200- 1600) / (5 - 8) = -800 per year. Since we know that the line crosses the point (5,4200), we can find the y-intercept, denoted as (b), using the formula y = mx + b.
Substitute m = -800, x = 5, and y = 4200 into the equation and solve for b
4200 = -800 * 5 + b
This simplifies to b= 4200 + 4000 = 8200
Therefore, the formula is P = -800t + 8200
To predict the moose population in 2003, simply substitute t = 13 into the formula (since 2003 is 13 years since 1990). Therefore,
P = -800 * 13 + 8200 = 2400
Therefore, the model predicts that the moose population in 2003 would be 2400.
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verify that sin2x=2cotsin^2x is an identity
Answer:
Vertify is an identity
Sin2x=2cotx(sin^2x)
starting from the right-hand side
2cotx(sin^2x)
=2(cosx/sinx)(sin^2x)
=2(cosx/sinx)(sin^2x)
=2sinxcosx=sin2x
ans:right-hand side=left-hand side
Step-by-step explanation:
Step-by-step explanation:
sin^2x = 2cotx sin^2x
Rewrite right side as fractions:
sin^2x = [tex]\frac{2}{1}[/tex] * [tex]\frac{cosx}{sinx}[/tex] * [tex]\frac{(sinx)(sinx)}{1}[/tex]
Multiply together [tex]\frac{cosx}{sinx}[/tex] and [tex]\frac{(sinx)(sinx)}{1}[/tex] :
sin^2x = [tex]\frac{2}{1}[/tex] * [tex]\frac{(cosx)(sinx)(sinx)}{sinx}[/tex]
Cancel out sinx on top and bottom:
sin^2x = [tex]\frac{2}{1}[/tex] * [tex]\frac{(sinx)(cosx)}{1}[/tex]
Multiply together 2 and (sinx)(cosx):
sin^2x = 2sinxcosx
Substitute sin^2x in for 2sinxcosx:
sin^2x = sin^2x
???????? Help me please
Answer:
D. (1, 9)
Step-by-step explanation:
Plug in each ordered pair into both inequalities. If both inequalities are satisfied, then the ordered pair is a solution.
The only ordered pair that works is (1, 9).
What is 5x times (3x^2 -5)
Answer:
[tex]\large\boxed{5x\times(3x^2-5)=15x^3-25x}[/tex]
Step-by-step explanation:
[tex]5x\times(3x^2-5)\qquad\text{use the distributive property:}\ a(b+c)=ab+ac\\\\=(5x)(3x^2)+(5x)(-5)\\\\=15x^3-25x[/tex]
The resulting product of the functions using the distributive property is
15x³ - 25x.
Product is an operation carried out when two or more variables, numbers, or functions are multiplied together.
Given the expression 5x(3x² - 5)
Taking the product:
5x(3x² - 5)
Expand using the distributive property
= 5x(3x²) - 5x(5)
= (5×3)(x × x²) - 25x
= 15x³ - 25x
Hence the resulting function is 15x³ - 25x.
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Jenny and Dan have $330 altogether. Jenny has $60 more than Dan. How much should Jenny take from Dan so that she has twice as much as Dan?
Answer:
$135
Step-by-step explanation:
Givens:
1) Jenny + Dan = $330
2) Jenny = $60 + Dan
Substitute Jenny's value into equation 1
(Dan = D)
$60 + D + D = $330
2D = $270
D = $135
Hope this helps :)
Find the perimeter of a parallelogram if two of its adjacent sides are 25 inches and 30 inches.
Answer:
P=110in
Step-by-step explanation:let me know if it's correct i'm not 100% sure
Answer:
110 inches
Step-by-step explanation:
The perimeter is basically the sum of all sides
In this case, it will be
25+25=50 for 2 opposite sides
30+30= 60 for 2 other opposite sides
hence 50+60= 110 inches
Evaluate the expression and place your answer in the space provided 3^2+(-2+3)•5
Answer:
14
Step-by-step explanation:
Evaluate exponents, followed by brackets, multiplication and addition
Given
3² + (- 2 + 3) × 5
= 9 + 1 × 5 ← exponents and bracket
= 9 + 5 ← multiplication
= 14 ← addition
Answer:
The correct answer is 14.
Step-by-step explanation:
I'll give you an a hint. You'd need to know about the order of operations is parenthesis, exponent, multiply, divide, add, and subtract.
First, do parenthesis.
(-2+3)=1
3²+1*5
Next, exponent.
3²=3*3=9
9+1*5
Then, multiply.
5*1=5
Finally, add.
9+5=14
So, the correct answer is 14.
I hope this helps!
A bag contains 5 blue marbles , 2 black marbles and 3 red marbles .a marble is randomly drawn from the bag the probability of not drawing a black marble is . The probability of drawing a red marble is
Answer: not a black marble: 4/5
Red marble: 3/10
Step-by-step explanation: Count the number of marbles.
5+2+3=10
The total number of marbles that aren’t black are 8 out of 10. The fraction is 8/10. It can be simplified to 4/5.
The number of red marbles is 3 out of 10. As a fraction, it’s 3/10.
solve 8x + 3y = 13 3x + 2y = 11 by using elimination. SHOW ALL WORK!! PLEASE HELP!!! THANK YOU SO MUCH!! :)))))
Answer:Y=7
Step-by-step explanation:
-8x-3y = -13 (eq 1)
-3x-2y = -11 (eq 2)
First, multiply equation 1 by 2.
-16x-6y = -26
Second, multiply equation 2 by 3.
-9x-6y = -33
Subtract the system of equations:
-16x-6y = -26
-9x-6y = -33
-7x = 7
x = -1
Substitute this value into one of the original equations to solve for y
-3x-2y = -11
-3(-1)-2y = -11
3-2y = -11
-2y = -14
y = 7
Really hope this helps :)
Hey There!
We have been given:
[tex]8x + 3y = 13 \\ 3x + 2y = 11[/tex]
Find the lcm of 3 and 2 to eliminate one equation:
3 * 2 = 6
2 * 3 = 6
Multiply each equation to get to 6:
[tex]2(8x + 3y = 13)\\ 16x + 6y = 26[/tex]
[tex]3(3x + 2y = 11)\\ 9x+6y=33[/tex]
Eliminate:
[tex]16x + 6y = 26\\ -(9x+6y=33)\\ -9x - 6y=-33[/tex]
Simplify:
[tex]16x + 6y = 26\\ -9x - 6y=-33 \\ 7x = -7[/tex]
Solve for x by dividing 7 in both sides:
[tex]7x = -7\\ x = -1[/tex]
Solve for y by substituting x in any equation with -1:
[tex]8(-1) + 3y = 13[/tex]
Simplify:
[tex]-8 + 3y = 13[/tex]
Add 8 in both sides:
[tex]3y = 21[/tex]
Solve for y by dividing 3 in both sides:
[tex]y = 7[/tex]
The value of x is -1 and the value of y is 7
Our answers:
x = -1
y = 7
What graph represents the compound inequality x<5/4 or x>5/2
For this case we have to:
[tex]x \leq \frac {5} {4}[/tex]: Represents all values less than or equal to[tex]\frac {5} {4}.[/tex]
[tex]x \geq \frac {5} {2}[/tex]: Represents all values greater than or equal to [tex]\frac {5} {2}.[/tex]
As the inequalities include the sign "=", then the borders of the graphs will be closed.
[tex]\frac {5} {4} = 1.25\\\frac {5} {2} = 2.5[/tex]
The word "or" indicates one solution or the other, so the correct option is graph B
ANswer:
Option B
Answer:
SECOND graph.
Step-by-step explanation:
Given compound inequality,
[tex]x \leq \frac{5}{4}\text{ or }x \geq \frac{5}{2}[/tex]
[tex]\because \frac{5}{4}=1.25\text{ or }\frac{5}{2}=2.5[/tex]
[tex]\implies x \leq 1.25\text{ or }x\geq 2.5[/tex]
If x ≥ 1.25
In the number line closed circle on 1.25 and shaded left side from 1.25,
If x ≤ 2.5
In the number line closed circle on 2.5 and shaded right side from 2.5
Hence, SECOND option is correct.
What is the area of parallelogram ABCD?
11 square units
13 square units
15 square units
16 square units
Answer:
13 square units
Step-by-step explanation:
First of all, you need to identify that ABCD is a rectangle (AB=CD and AD=BC).
The area of a rectangle is calculated by multiplying the length and the width.
Secondly, we use the Pythagoras’s theorem to calculate side CD and AD (the length and width). I’ve added some labels to your original diagram (see picture attached) so that it’s easier to understand.
The Pythagoras’s theorem is a^2 + b^2 = c^2 (c is the hypotenuse).
So, for side CD:
3^2 + 1^2 = (CD)^2
9 + 1 = (CD)^2
CD = √ 10
and for side AD:
4^2 + 1^2 = (AD)^2
16 + 1 = (AD)^2
AD = √17
Lastly, to calculate the area:
√10 x √17 = 13.04
Your answer is 13 square units.
Hope this helped :)
Answer:
Option B. 13 square units
Step-by-step explanation:
Area of a parallelogram is defined by the expression
A = [tex]\frac{1}{2}(\text{Sum of two parallel sides)}[/tex] × (Disatance between them)
Vertices of A, B, C and D are (3, 6), (6, 5), (5, 1) and (2, 2) respectively.
Length of AB = [tex]\sqrt{(x-x')^{2}+(y-y')^{2}}[/tex]
= [tex]\sqrt{(5-6)^{2}+(6-3)^{2}}[/tex]
= [tex]\sqrt{10}[/tex]
Since length of opposite sides of a parallelogram are equal therefore, length of CD will be same as [tex]\sqrt{10}[/tex]
Now we have to find the length of perpendicular drawn on side AB from point D or distance between parallel sides AB and CD.
Expression for the length of the perpendicular will be = [tex]\frac{|Ax_{1}+By_{1}+C|}{\sqrt{A^{2}+B^{2}}}[/tex]
Slope of line AB (m) = [tex]\frac{y-y'}{x-x'}[/tex]
= [tex]\frac{6-5}{3-6}=-(\frac{1}{3} )[/tex]
Now equation of AB will be,
y - y' = m(x - x')
y - 6 = [tex]-\frac{1}{3}(x-3)[/tex]
3y - 18 = -(x - 3)
3y + x - 18 - 3 = 0
x + 3y - 21 = 0
Length of a perpendicular from D to side AB will be
= [tex]\frac{|(2+6-21)|}{\sqrt{1^{2}+3^{2}}}[/tex]
= [tex]\frac{13}{\sqrt{10}}[/tex]
Area of parallelogram ABCD = [tex]\frac{1}{2}(AB+CD)\times (\text{Distance between AB and CD})[/tex]
= [tex]\frac{1}{2}(\sqrt{10}+\sqrt{10})\times (\frac{13}{\sqrt{10} } )[/tex]
= [tex]\sqrt{10}\times \frac{13}{\sqrt{10} }[/tex]
= 13 square units
Option B. 13 units will be the answer.
In the formula for average rate of change, what does the triangle in front of the x and y stand for?
Answer:
It a delta notation that change in y over change in x
Answer:
C
Step-by-step explanation:
Edge 2021
If f(x)=x2+3x+5, what is f(a+h)?
Answer:
[tex]\large\boxed{D.\ a^2+2ah+h^2+3a+3h+5}[/tex]
Step-by-step explanation:
[tex]f(x)=x^2+3x+5\\\\f(a+h)\to\text{exchange x to (a + h)}:\\\\f(a+h)=(a+h)^2+3(a+h)+5\\\\\text{use}\ (a+b)^2=a^2+2ab+b^2\ \text{and the distributive property}\\\\f(a+h)=a^2+2ah+h^2+3a+3h+5[/tex]
how to find perimeter of ceiling
Answer:
add up the lengths of all the sides of the ceiling,is there a diagram that comes with this or something?
Step-by-step explanation:
Answer: multiply
Step-by-step explanation: you must multiply length x width
Solve x^2-8x=3 by completing the square. Which is the solution set of the equation
Answer:
{-0.36, 8.36) to the nearest hundredth.
Step-by-step explanation:
x^2 - 8x = 3
(x - 4)^2 - 16 = 3
(x - 4)^2 = 19
Taking square roots:
x - 4 = +/- √19
x = 4 +/- √19
x = {-0.36, 8.36} to nearest 1/100.
For this case we have the following expression:
[tex]x ^ 2-8x = 3[/tex]
We must complete squares.
So:
We divide the middle term between two and we square it:
[tex](\frac {-8} {2}) ^ 2[/tex], then:
[tex]x ^ 2-8x + (\frac {-8} {2}) ^ 2 = 3 + (\frac {-8} {2}) ^ 2\\x ^ 2-8x + (- 4) ^ 2 = 3 + 16[/tex]
We have to, by definition:
[tex](a-b) ^ 2 = a ^ 2-2ab + b ^ 2[/tex]
Then, rewriting:
([tex](x-4) ^ 2 = 19[/tex]
To find the roots, we apply square root on both sides:
[tex]x-4 = \sqrt {19}[/tex]
We have two solutions:
[tex]x_ {1} = \sqrt {19} +4\\x_ {2} = - \sqrt {19} +4[/tex]
Answer:
([tex](x-4) ^ 2 = 19\\x_ {1} = \sqrt {19} +4\\x_ {2} = - \sqrt {19} +4[/tex]
Given the function f(x)=1+5x^2, calculate the following values:
f(a)=
f(a+h)=
f(a+h)−f(a)/h=
Answer:
[tex]f(a)=1+5a^2[/tex]
[tex]f(a+h)=1+5a^2+10ah+5h^2[/tex]
[tex]\frac{f(a+h)-f(a)}{h}=10a+5h[/tex]
Step-by-step explanation:
We are given [tex]f(x)=1+5x^2[/tex].
Find [tex]f(a)[/tex]. All this means is replace [tex]x[/tex] in [tex]f(x)=1+5x^2[/tex] with [tex]a[/tex].
[tex]f(x)=1+5x^2[/tex]
[tex]f(a)=1+5a^2[/tex]
Find [tex]f(a+h)[/tex]. All this means is replace [tex](a+h)[/tex] in [tex]f(x)=1+5x^2[/tex] with [tex](a+h)[/tex].
[tex]f(x)=1+5x^2[/tex]
[tex]f(a+h)=1+5(a+h)^2[/tex]
[tex]f(a+h)=1+5(a+h)(a+h)[/tex]
[tex]f(a+h)=1+5(a^2+2ah+h^2)[/tex]
[tex]f(a+h)=1+5a^2+10ah+5h^2[/tex]
Find [tex]\frac{f(a+h)-f(a)}{h}[/tex]. So we got to put some parts together; the parts above:
[tex]\frac{f(a+h)-f(a)}{h}[/tex]
[tex]\frac{(1+5a^2+10ah+5h^2)-(1+5a^2)}{h}[/tex]
Now in the first ( ) I see 1+5a^2 and in the second ( ) I see 1+5a^2, so this means you have (1+5a^2)-(1+5a^2) which equals 0.
[tex]\frac{10ah+5h^2}{h}[/tex]
Now assuming h is not 0. we can divide top and bottom by h.
[tex]\frac{10a+5h}{1}[/tex]
[tex]10a+5h[/tex]
To calculate the values for the quadratic function f(x) = 1 + 5x^2, substitute the necessary values for x. The difference quotient involves a simplification process of the terms after substitution.
The problem is to evaluate the function f(x) = 1 + 5x^2 at a given value a and at a + h, and then to find the difference quotient which is part of the process to find the derivative of the function. First, we calculate f(a), then f(a + h), and lastly the difference quotient f(a + h) - f(a) / h.
To get f(a), we substitute x with a in the function, so we get:
f(a) = 1 + 5a^2
For f(a + h), we substitute x with (a + h):
f(a + h) = 1 + 5(a + h)^2
Now to find the difference quotient (f(a + h) - f(a)) / h:
(f(a + h) - f(a)) / h = (1 + 5(a + h)^2 - (1 + 5a^2)) / h
We can simplify this further by expanding and combining like terms, eventually canceling out the h in the denominator. However, without further expansion and simplification, the expression is already accurate.
Find the area of the trapezoid.
Answer:
[tex]\large\boxed{A=54\ m^2}[/tex]
Step-by-step explanation:
The formula of an area of a trapezoid:
[tex]A=\dfrac{b_1+b_2}{2}\cdor h[/tex]
b₁, b₂ - bases
h - height
We must use the Pythagorean theorem:
[tex]x^2+8^2=10^2[/tex]
[tex]x^2+64=100[/tex] subtract 64 from both sides
[tex]x^2=36\to x=\sqrt{36}\\\\x=6\ m[/tex]
We have b₁ = 6 + 6 = 12m, b₂ = 6m and h = 8m.
Substitute:
[tex]A=\dfrac{12+6}{2}\cdot6=\dfrac{18}{2}\cdot6=(9)(6)=54\ m^2[/tex]