Answer:
4.9 × 10^1
hopefully this helped :3
Answer:
4.9 * 10 ^1
Step-by-step explanation:
We want one digit to the left of the decimal
49
4.9
We moved the decimal one place to the left
One is the exponent. Since we moved to the left the exponent is positive
4.9 * 10 ^1
Am I correct? (you don't need to show work but you need to tell me the answer) But the person with the most work will receive the brainliest!
━━━━━━━☆☆━━━━━━━
▹ Answer
Exponential
▹ Step-by-Step Explanation
Every time 1 is added to x, y multiples by 2.
0 + 1 = 1
1 + 1 = 2
2 + 1 = 3
3 + 1 = 4
-2 * 2 = -4
-4 * 2 = -8
-8 * 2 = -16
-16 * 2 = -32
Hope this helps!
CloutAnswers ❁
Brainliest is greatly appreciated!
━━━━━━━☆☆━━━━━━━
which is equivalent to 243 Superscript two-fifths?
Answer:
9
i got it right on my test
The equivalent expressions to 243 Superscript two-fifths will be 9.
What are equivalent expressions?Those expressions who might look different but their simplified forms are same expressions are called equivalent expressions.
To derive equivalent expressions of some expression, we can either make it look more complex or simple, we simplify it.
Simplification usually involves making the expression simple and easy to use later,
The given expression is;
[tex]243^{2/5}[/tex]
We know that 2/3 = 0.4
So, the equivalent expressions to the given function;
[tex]243^{0.4}\\\\= 9[/tex]
Thus, The equivalent expressions to 243 Superscript two-fifths will be 9.
Learn more about expression here;
https://brainly.com/question/14083225
#SPJ2
What is the inverse of the function f(x) = 2x + 1?
1
1
h(x) =
X-
2
2
1
1
Oh(x) =
- x +
O h(x) =
3x-2
Oh(x) =
= {x+2
Mark this and return
Save and Exit
Next
Submit
Answer:
[tex]f^{-1} = \frac{x-1}{2}[/tex]
Step-by-step explanation:
[tex]f(x) = 2x+1[/tex]
Replace it with y
[tex]y = 2x+1[/tex]
Exchange the values of x and y
[tex]x = 2y+1[/tex]
Solve for y
[tex]x = 2y+1[/tex]
Subtracting 1 from both sides
[tex]2y = x-1[/tex]
Dividing both sides by 2
[tex]y = \frac{x-1}{2}[/tex]
Replace it by [tex]f^{-1}[/tex]
So,
[tex]f^{-1} = \frac{x-1}{2}[/tex]
Answer:
[tex]\displaystyle f^{-1}(x)= \frac{1}{2}x - \frac{1}{2}[/tex]
Step-by-step explanation:
f(x) = 2x + 1
f(x) = y (output)
y = 2x + 1
Solve for x.
y - 1 = 2x
Divide 2 on both sides.
y/2 - 1/2 = x
1/2y - 1/2 = x
Switch variables.
1/2x - 1/2 = y
[tex]f^{-1}(x)= \frac{1}{2}x - \frac{1}{2}[/tex]
united services and supplies reports net income of $100000 and cost of goods of $353000. If US&S's gross profit rate was 30%, net sales were
Answer:
Net sales is $ 504285.71
Step-by-step explanation:
We have the following:
Let net sales be x.
Net sales - Cost of goods sold = Gross profit
We replace and we are left with:
x - $ 353000 = x * 30%
x - $ 353000 = 0.30 * x
x - 0.30 * x = $ 353000
0.7 * x = $ 353000
x = $ 353000 / 0.7
x = $ 504285.71
Therefore, net sales is $ 504285.71
Consider country Z with a GDP level of 210000 and a growth rate of 5% in 2019 (ie calculated at the end of year 2019). The experts predict that the growth of the economy of country Z will gradually slowdown in the coming year. More precisely, they foresee the following growth rate for the future: 2019-2022 (5%), 2022-2025 (3%).
a. Assuming that the prediction of the experts listed above are accurate, when in the future will country Z's GDP double compared to the GDP level of 2019??
b. What would country Z's GDP growth rate be from 2025 and so on at 1%. Explain your reasoning carefully
Answer:
a. Country Z's GDP will double in 23 years' time, given the experts prediction of 5% growth for 3 and 3% thereafter.
b. Z's GDP 's growth rate would be 39%.
This growth rate is calculated by determining Z's GDP from the end of 2025 as 292,000. So, (292,000 - 210,000)/210,000 x 100 = 39% from the 2019 base year.
Step-by-step explanation:
a) 2019 Country Z's GDP = 210,000
2019 Growth rate = 5%
Future growth rate:
2019 - 2022 = 5%
2022- 2025 = 3%
2025 - so on = 1%
Let Country Z's GDP in 2019 = G₀ = 210,000
n = number of years from 2019
g = growth rate = 5%
(1 + g)ⁿ = increase in GDP as a result of the growth rate and number of years
Gⁿ = GDP in n years
Therefore, Gⁿ = G₀(1 + g)ⁿ
b) With GDP growth of 5% from 2019 to 2022, the GDP will be
= 210,000 (1 + 5%)³
= 210,000 x 1.158
= 243,000 approx.
c) From 2022 to 2025 at 3%, the GDP will be
= 243,000 (1 + 3%)∧20
= 243,000 x 1.817
= 441,531
d) Z's GDP from 2025 with 1% growth and so on, will become double at:
Gⁿ = 265,600(1 + 1%)∧48
= 265,600 x 1.61
= 428,000
48 + 6 = 54 years.
Richard works at an ice cream shop. Regular cones get two scoops of ice cream; large cones get three scoops. One hot Saturday, Richard scooped 234 regular cones and 156 large cones. One scoop of ice cream is 3 ounces. A tub of ice cream weighs 10 pounds. How many tubs of ice cream did Richard use to make the cones? 3. Draw a picture or a chart that shows the information and the question A. 936 tubs B. 468 tubs C. 175 1/2 tubs D. 17 11/20 tubs
Answer:
D. 17 11/20 tubs
Step-by-step explanation:
Convert the cones into scoops.
234 regular cones = 468 scoops; 156 large cones = 468
So in total, Richard scooped 936 scoops. Since each scoop is 3 oz:
936 scoops * 3 oz = 2808 oz
Because a tub is 10 pounds or 160 oz:
2808 / 160 = 17.55 tubs
The answer is D. Here is how I found the answer:
First, figure out how many scoops were served. There were 234 Regular cones with 2 scoops each and 156 large cones with 3 each. So we do 234×2 + 156×3 = 936 scoops.Next, we need to know how many ounces of icecream were served. Since 936 scoops were served and each scoop is 3 ounces, we do 936×3=2808 ounces.Then we need to go from ounces to pounds. Since there is 16 ounces in a pound, we simply do 2808/16 = 175.5 pounds.Finally, we go from pounds to tubs of ice cream. Each tub is 10 pounds, so we'll do 175.5/10 = 17.55, or 17 11/20 tubs.Hope this helps!
someone please help me!!!
Explanation:
Surface area of a cone = pi*r^2 + pi*r*sqrt(r^2+h^2)
r = radius
h = height of cone
In this case,
r = 8 is the radius
h = 41
So,
SA = surface area
SA = pi*r^2 + pi*r*sqrt(r^2+h^2)
SA = pi*8^2 + pi*8*sqrt(8^2+41^2)
SA = 1250.936884057 use a calculator for this step
SA = 1251 square meters approximately
Eighty-five percent of the people that use the Internet order something online. Find the probability that only 7 of 10 Internet users will order something online
Answer:
the probability that 7 out 10 Internet users will order something online is 0.78 or 78%
Step-by-step explanation:
Given that:
Internet users who order something online = 85%
To find:
Probability that 7 out of 10 Internet users will order something online ?
Solution:
This problem is of binomial distribution of probability.
where, p = 0.85
q = 1-p = 0.15
r = 7 and
n = 10
Formula is given as:
[tex]P(x = r) = _nC_r q^{n-r}p^r[/tex]
Putting all the values:
[tex]P(x = 7) = _{10}C_7 0.15^{10-7}\times 0.85^7\\P(x = 7) = _{10}C_3 0.15^{3}\times 0.85^7\\P(x = 7) = 10\times 9 \times 8 \times 0.15^{3}\times0.85^7\\P(x = 7) = 720 \times .0034 \times0.32\\P(x = 7) = 0.78[/tex]
So, the probability that 7 out 10 Internet users will order something online is 0.78 or 78%.
Each year, more than 2 million people in the United States become infected with bacteria that are resistant to antibiotics. In particular, the Centers of Disease Control and Prevention have launched studies of drug-resistant gonorrhea.† Suppose that, of 174 cases tested in a certain state, 11 were found to be drug-resistant. Suppose also that, of 375 cases tested in another state, 7 were found to be drug-resistant. Do these data suggest a statistically significant difference between the proportions of drug-resistant cases in the two states? Use a 0.02 level of significance. (Let p1 = the population proportion of drug-resistant cases in the first state, and let p2 = the population proportion of drug resistant cases in the second state).
A. State the null and alternative hypotheses.
B. Find the value of the test statistic.
C. What is the p-value?
D. What is your conclusion?
1. Reject H0. There is a significant difference in drug resistance between the two states.
2. Do not reject H0. There is a significant difference in drug resistance between the two states.
3. Reject H0. There is not a significant difference in drug resistance between the two states.
4. Do not reject H0. There is not a significant difference in drug resistance between the two states.
Answer:
A)
Null hypothesis:H₀:- There is no significant difference between in drug resistance between the two states
Alternative Hypothesis :H₁:
There is significant difference between in drug resistance between the two states
B)
The calculated value Z = 2.7261 > 2.054 at 0.02 level of significance
Rejected H₀
There is a significant difference in drug resistance between the two states.
C)
P - value = 0.0066
P - value = 0.0066 < 0.02
D)
1) Reject H₀
There is a significant difference in drug resistance between the two states.
Step-by-step explanation:
Step(i):-
Given first sample size n₁ = 174
Suppose that, of 174 cases tested in a certain state, 11 were found to be drug-resistant.
First sample proportion
[tex]p_{1} = \frac{x_{1} }{n_{1} } = \frac{11}{174} = 0.0632[/tex]
Given second sample size n₂ = 375
Given data Suppose also that, of 375 cases tested in another state, 7 were found to be drug-resistant
Second sample proportion
[tex]p_{2} = \frac{x_{2} }{n_{2} } = \frac{7}{375} = 0.0186[/tex]
Step(ii):-
Null hypothesis:H₀:- There is no significant difference between in drug resistance between the two states
Alternative Hypothesis :H₁:
There is significant difference between in drug resistance between the two states
Test statistic
[tex]Z = \frac{p_{1}-p_{2} }{\sqrt{PQ(\frac{1}{n_{1} }+\frac{1}{n_{2} } ) } }[/tex]
Where
[tex]P = \frac{n_{1}p_{1} +n_{2} p_{2} }{n_{1} +n_{2} }[/tex]
[tex]P = \frac{174 (0.0632) + 375 (0.0186) }{174+375 } = \frac{17.9718}{549} = 0.0327[/tex]
Q = 1 - P = 1 - 0.0327 = 0.9673
Step(iii):-
Test statistic
[tex]Z = \frac{p_{1}-p_{2} }{\sqrt{PQ(\frac{1}{n_{1} }+\frac{1}{n_{2} } ) } }[/tex]
[tex]Z = \frac{0.0632-0.0186 }{\sqrt{0.0327 X 0.9673(\frac{1}{174 }+\frac{1}{375 } ) } }[/tex]
Z = 2.7261
Level of significance = 0.02 or 0.98
The z-value = 2.054
The calculated value Z = 2.7261 > 2.054 at 0.02 level of significance
Reject H₀
There is a significant difference in drug resistance between the two states.
P- value
P( Z > 2.7261) = 1 - P( Z < 2.726)
= 1 - ( 0.5 + A (2.72))
= 0.5 - 0.4967
= 0.0033
we will use two tailed test
2 P( Z > 2.7261) = 2 × 0.0033
= 0.0066
P - value = 0.0066 < 0.02
Reject H₀
There is a significant difference in drug resistance between the two states.
PLEASE AWNSER SOON!!! IM ALMOST OUT OF TIME!!!!!!!Rectangle JKLM is rotated 90° clockwise about the origin. On a coordinate plane, rectangle J K L M has points (negative 4, 1), (negative 1, 1), (negative 1, negative 1), (negative 4, negative 1). What are the coordinates of J’? J’(–1, –4) J’(4, –1) J’(1, 4) J’(4, 1)
Answer:
The answer is C (j' 1,4). It is not B J'(4, 1) because the new coordinate is moving 90 degrees clockwise not 180 degrees.
Step-by-step explanation:
J' will have coordinates (1, 4) after 90° clockwise rotation
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
Rectangle JKLM is rotated 90° clockwise about the origin.
On a coordinate plane, rectangle J K L M has points (-4, 1), (-1, 1), (- 1, - 1), (- 4, - 1).
After rotating the rectangle 90° clockwise about the origin, the point J(-4, 1) will be transformed to a new point J' with coordinates (y, -x).
Thus, J' will have coordinates (1, 4).
Therefore, J' will have coordinates (1, 4) after 90° clockwise rotation
To learn more on Graph click:
https://brainly.com/question/17267403
#SPJ3
Please Help ASAP! Consider the function below.
f(x) = 2X - 2.
Which of these graphs represent the inverse of the function F??
Answer:
1st Graph
Step-by-step explanation:
Step 1: Find the inverse
y = 2x - 2
x = 2y - 2
x + 2 = 2y
y = (x + 2)/2
Step 2: Graph the inverse
Simply use a graphing calculator and we should see our answer is the top left graph.
3. Compare corresponding sides that include the
corresponding, congruent angles to show they are in
proportion.
Help please
Answer:
AB/AD=1/2 AC/AE=1/2
Step-by-step explanation:
The distance between 2 points A(x1; y1) and B(x2; y2) can be calculated as:
AB= sqrt((x1-x2)²+(y1-y2)²)
So in our case AB= sqrt ((1-3)²+(4-1)²)= sqrt(2²+3²)=sqrt(13)
AD=sqrt ((-1-3)²+(7-1)²)= sqrt(4²+6²)=sqrt(52)
AB/AD=sqrt(13)/sqrt(52)= sqrt(13/52)=sqrt(1/4) =1/2
AB/AD =1/2
AC= sqrt ((5-3)²+(3-1)²)= sqrt(2²+2²)=sqrt(8)
AE=sqrt ((7-3)²+(5-1)²)= sqrt(4²+4²)=sqrt(32)
AC/AE=sqrt(8)/sqrt(32)= sqrt(8/32)=sqrt(1/4)=1/2
AC/AE=1/2
Answer:
first one is 1/2
second one is 1/2
Step-by-step explanation:
got it right on edge 2020
What is the equation of the graphed line written in
standard form?
O 2x - y = -4
O 2x - y = 4
O y = 2x – 4
O y=x-4
Answer:
2x-y=4
Step-by-step explanation:
Standard form of a line: Ax+by=c
Use slope intercept form: y=mx+b
slope= 2
y=2x-4
Add 4 to both sides.
y+4=2x
subtract y from both sides.
4=2x-y
Rotate the equation
2x-y=4
Answer:
2x-y=4
Step-by-step explanation:
y=2x-4 is the slope intercept.
y-2x=-4
-2x+y=-4
2x-y=4
the class mean was 72 with a standard deviation of 4.2. Calculate the z-score (to 2 decimal places) for a person who received score of 59. Is it usual or unusual?
lllluvyvyvyvtctctvtvtctvtctctctctctctctvtvyvyvyvyvyvyvtvtv
I NEED HELP PLEASE, THANKS! :)
Write 18(cos169° + isin169°) in rectangular form. Round numerical entries in the answer to two decimal places. (Show work)
Answer:
-17.67 +3.43i
Step-by-step explanation:
Carry out the indicated math:
18 cis 169° = (18·cos(169°) +i·18·sin(169°)) = (18·(-0.9816) +i·18·0.1908)
= -17.67 +i·3.43
Answer:
The rectangular form is z = -17.67 + i 3.43
Step-by-step explanation:
Plz answer what is in the screen shot!
Answer:
([tex]\sqrt{15}[/tex])/7
Step-by-step explanation:
Let b be the tird side of the triangle
tanθ= b/c
using the pythagorian theorem we get :
a²+b²= c² ⇒ b²= c²-a²= 8²-7²=15 ⇒b=√15
so: tanθ= √15/7
write a polar equation of a conic with the focus at the origin and the given data.hyperbola, eccentricity 3.5, directrix y
Complete Question is;
Write a polar equation of a conic with the focus at the origin and the given data. hyperbola, eccentricity 3.5, directrix y =2
Answer:
r = 14/(2 + 7(sinθ))
Step-by-step explanation:
We are given;
Eccentricity;e = 3.5
Directrix; y = 2
This means that d = 2
We are told that the focus is at the origin, so since the directrix is at y = 2,then the part of the hyperbola that is closest to this focus will open downwards and the equation is given by;
r = ed/(1 + e•sinθ)
Plugging in the relevant values, we have;
r = (3.5 × 2)/(1 + 3.5(sinθ))
r = 7/(1 + 3.5(sinθ))
To make every figure a whole number, let's multiply numerator and denominator by 2 to give;
r = 14/(2 + 7(sinθ))
The side, s, of a square with area A square feet is given by the formula s = square root A. Find the perimeter of a square with an area of 36 square feet. ______________ ft
Answer:
24 feet.
Step-by-step explanation:
The side of the square = √36 = 6 feet.
The square has 4 equal sides so the perimeter = 4*6 = 24 feet.
Fill in the missing information. Tim Worker is doing his budget. He discovers that the average miscellaneous expense is $45.00 with a standard deviation of $16.00. What percent of his expense in this category would he expect to fall between $38.60 and $57.80?
Answer:
[tex] P(38.6 <X <57.8)[/tex]
And we can assume a normal distribution and then we can solve the problem with the z score formula given by:
[tex]z=\frac{X -\mu}{\sigma}[/tex]
And replacing we got:
[tex] z=\frac{38.6- 45}{16}= -0.4[/tex]
[tex] z=\frac{57.8- 45}{16}= 0.8[/tex]
We can find the probability of interest using the normal standard table and with the following difference:
[tex] P(-0.4 <z<0.8)= P(z<0.8) -P(z<-0.4) = 0.788-0.345= 0.443[/tex]
Step-by-step explanation:
Let X the random variable who represent the expense and we assume the following parameters:
[tex]\mu = 45, \sigma 16[/tex]
And for this case we want to find the percent of his expense between 38.6 and 57.8 so we want this probability:
[tex] P(38.6 <X <57.8)[/tex]
And we can assume a normal distribution and then we can solve the problem with the z score formula given by:
[tex]z=\frac{X -\mu}{\sigma}[/tex]
And replacing we got:
[tex] z=\frac{38.6- 45}{16}= -0.4[/tex]
[tex] z=\frac{57.8- 45}{16}= 0.8[/tex]
We can find the probability of interest using the normal standard table and with the following difference:
[tex] P(-0.4 <z<0.8)= P(z<0.8) -P(z<-0.4) = 0.788-0.345= 0.443[/tex]
How can you solve for x in the proportion StartFraction 7 over 8 EndFraction = StartFraction x over 24 EndFraction? Set the sum of 7 and 8 equal to the sum of 24 and x, and then solve for x. Set the sum of 7 and 24 equal to the sum of 8 and x, and then solve for x. Set the product of 7 and 8 equal to the product of 24 and x, and then solve for x. Set the product of 7 and 24 equal to the product of 8 and x, and then solve for x.
Answer: the last one
Step-by-step explanation: because 7 times 24 equals 168 and x would be 21 so 21 times 8 equals 168.
Answer: The answer is D
Step-by-step explanation:
What steps would you take to determine if these figures are similar? Check all that apply. Use a scale factor of 2. Multiply the vertices of polygon ABCD by One-half. Translate the intermediate image 4 units down. Perform two different dilations. Reflect the intermediate image.
Answer:
Well I took it"s Reflect the intermediate image. and Multiply the vertices of polygon ABCD by One-half.
Step-by-step explanation:
Answer:
2 and 5 or B and E
Step-by-step explanation:
i did it on edge! ; )
If f(x) = 7 + 4x and g (x) = StartFraction 1 Over 2 x EndFraction, what is the value of (StartFraction f Over g EndFraction) (5)?
Answer:
270
Step-by-step explanation:
f(5) = 7 +4·5 = 27
g(5) = 1/(2·5) = 1/10
The ratio of functions is the ratio of their individual values:
(f/g)(5) = f(5)/g(5) = 27/(1/10)
(f/g)(5) = 270
Which expression is equivalent to 8 - (6r + 2)? A. -6r + 6 B. 2r + 2 C. 6r + 10 D. -6r + 10
Answer:
6 -6r
Step-by-step explanation:
8 - (6r + 2)
Distribute the minus sign
8 - 6r -2
Combine like terms
6 -6r
Answer:
A. -6r+6
Step-by-step explanation:
We are given the expression:
8-(6r+2)
First, let's distribute the negative sign. Multiply each term inside the parentheses by -1.
8+ (-1*6r) + (-1*2)
8-6r-2
Now, combine like terms. Add the constants, or terms without a variable.
-6r + (8+-2)
-6r + (8-2)
-6r + (6)
-6r+6
The answer is A. -6r+6
-9 ≤ -7 true or false?
Which correctly describes how to determine the measure of angle 1? 2 parallel lines are crossed by a transversal to form 8 angles. Clockwise from top left, the angles are 1, 2, blank, blank; blank 81 degrees, blank, blank. The 81° angle and angle 2 are corresponding angles so angle 2 must measure 81°. Angles 1 and 2 are supplementary angles so angle 1 must measure 99°. The 81° angle and angle 2 are alternate interior angles so angle 2 must measure 81°. Angles 1 and 2 are supplementary angles so angle 1 must measure 99°. The 81° angle and angle 2 are corresponding angles so angle 2 must measure 99°. Angles 1 and 2 are supplementary angles so angle 1 must measure 81°. The 81° angle and angle 2 are alternate interior angles so angle 2 must measure 99°. Angles 1 and 2 are supplementary angles so angle 1 must measure 81°.
Answer:
m∠1 = 99°
Step-by-step explanation:
∠2 and ∠81° are Corresponding Angles, so they are equal to each other:
∠2 = 81°
∠1 and ∠2 are Supplementary Angles, so they add up to 180°:
180 - m∠2 = m∠1
180 - 81 = 99°
Sandy is working with a carpenter to frame a house. They are using 8-foot-long boards, but each board must be cut to be 7 feet inches long. How much is cut off each board? Note: 1 foot = 12 inches A. 1 1/4 inches B. 1 3/4 inches C. 7 inches D. 3/4 inch
Answer:
c
Step-by-step explanation:
plato
Kiemanh is solving the equation 15 r minus 6 r = 36. What is the value of r?
━━━━━━━☆☆━━━━━━━
▹ Answer
r = 4
▹ Step-by-Step Explanation
15r - 6r = 36
Collect like terms
9r = 36
Divide
r = 4
Final Answer
r = 4
Hope this helps!
- CloutAnswers ❁
Brainliest is greatly appreciated!
━━━━━━━☆☆━━━━━━━
Answer: The answer would be 4, which is B, hopefully this helps.
Prove the formula for (d/dx)(cos−1(x)) by the same method as for (d/dx)(sin−1(x)). Let y = cos−1(x). Then cos(y) = and 0 ≤ y ≤ π ⇒ −sin(y) dy dx = 1 ⇒
Answer:
[tex]\frac{d(cos^{-1}x )}{dx} = \frac{-1}{\sqrt{1-x^2} }[/tex]
Step-by-step explanation:
Given the differential (d/dx)(cos−1(x)), to find the equivalent formula we will differentiate the inverse function using chain rule as shown below;
let;
[tex]y = cos^{-1} x \\\\taking \ cos\ of\ both\ sides\\\\cosy = cos(cos^{-1} x)\\\\cosy = x\\\\x = cosy\\\\\frac{dx}{dy} = -siny\\[/tex]
[tex]\frac{dy}{dx} = \frac{-1}{sin y} \\\\from\ trigonometry\ identity,\ sin^{2} x+cos^{2}x = 1\\sinx = \sqrt{1-cos^{2} x}[/tex]
Therefore;
[tex]\frac{dy}{dx} = \frac{-1}{\sqrt{1-cos^{2}y } }[/tex]
Since x = cos y from the above substitute;
[tex]\frac{dy}{dx} = \frac{-1}{\sqrt{1-x^{2}} }[/tex]
Hence, [tex]\frac{d(cos^{-1}x )}{dx} = \frac{-1}{\sqrt{1-x^2} }[/tex] gives the required proof
Evaluate the following integral using trigonometric substitution.
Integral from 7 StartRoot 49 - x2 EndRoot dx
1. What substitution will be the most helpful for evaluating thisintegral?
2. Find dx?
3. Rewrite the given integral using substitution.
Answer:
Step-by-step explanation:
1. Given the integral function [tex]\int\limits {\sqrt{a^{2} -x^{2} } } \, dx[/tex], using trigonometric substitution, the substitution that will be most helpful in this case is substituting x as [tex]asin \theta[/tex] i.e [tex]x = a sin\theta[/tex].
All integrals in the form [tex]\int\limits {\sqrt{a^{2} -x^{2} } } \, dx[/tex] are always evaluated using the substitute given where 'a' is any constant.
From the given integral, [tex]\int\limits {7\sqrt{49-x^{2} } } \, dx = \int\limits {7\sqrt{7^{2} -x^{2} } } \, dx[/tex] where a = 7 in this case.
The substitute will therefore be [tex]x = 7 sin\theta[/tex]
2.) Given [tex]x = 7 sin\theta[/tex]
[tex]\frac{dx}{d \theta} = 7cos \theta[/tex]
cross multiplying
[tex]dx = 7cos\theta d\theta[/tex]
3.) Rewriting the given integral using the substiution will result into;
[tex]\int\limits {7\sqrt{49-x^{2} } } \, dx \\= \int\limits {7\sqrt{7^{2} -x^{2} } } \, dx\\= \int\limits {7\sqrt{7^{2} -(7sin\theta)^{2} } } \, dx\\= \int\limits {7\sqrt{7^{2} -49sin^{2}\theta } } \, dx\\= \int\limits {7\sqrt{49(1-sin^{2}\theta)} } } \, dx\\= \int\limits {7\sqrt{49(cos^{2}\theta)} } } \, dx\\since\ dx = 7cos\theta d\theta\\= \int\limits {7\sqrt{49(cos^{2}\theta)} } } \, 7cos\theta d\theta\\= \int\limits {7\{7(cos\theta)} }}} \, 7cos\theta d\theta\\[/tex]
[tex]= \int\limits343 cos^{2} \theta \, d\theta[/tex]
Circle X with a radius of 6 units and circle Y with a radius of 2 units are shown. Which steps would prove the circles similar? Translate the circles so they share a common center point, and dilate circle Y by a scale factor of 4. Translate the circles so the center of one circle rests on the edge of the other circle, and dilate circle Y by a scale factor of 4. Translate the circles so they share a common center point, and dilate circle Y by a scale factor of 3. Translate the circles so the center of one circle rests on the edge of the other circle, and dilate circle Y by a scale factor of 3.
Answer:
See bolded below.
Step-by-step explanation:
Take a look at the attachment below. It represents circle x and y, with respect to each of their radii. In each of the options we are given, we would have to translate and dilate the circles, by a fixed scale factor -
Now the first thing one would do is translate the circles so that they share a common center point, or in other words the center of one circle rests on the edge of the other circle. That way when dilating circle y, it may fit into circle x as it expands.
The second point is how much this smaller circle ( circle y ) has to expand. The radius of circle y being 2, has to increase by 3 times the value to equal the radius of circle x, and hence has to dilate by a scale factor of 3 as to match circle x,
Solution = " Translate the circles so the center of one circle rests on the edge of the other circle, and dilate circle Y by a scale factor of 3 " / Option D