The area of a circle is changing at a rate of 576π cm²/sec when the radius of the circle is 32 cm.
What is termed as the Rate of Change?The derivative can be thought of as a change rate. If two variables are related, the derivative of the initial variable to respect to the second variable can be used to calculate the rate of change of one of the variation with respect to the other variable.For the given question;
Express the circle's area as a function of the its radius.
A = πr²
Determine the area's derivative with respect towards its radius.
dA/dr = 2πr
Using the chain rule, calculate the derivative of a area with respect to time.
dA/dt = dA/dr . dr/dt
dA/dt = 2πr(9)
dA/dt = 18πr
When the radius is 32 cm, find the derivative of a area with respect to time.
dA/dt = 18π(32)
dA/dt = 576π
As a result, the area of a circle is changing at a rate of 576π cm²/sec when the radius of the circle is 32 cm.
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i'm still confused. helpppppppppppppppppppppppppppppppppppppppppppppppppppppp meeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Answer:
I believe you forgot to add the attachment
Step-by-step explanation:
There is no math problem I can see to help you
Which exponential equation is equivalent to this logarithmic equation? \log _(5)x - \log _(5)25=7
The exponential equation [tex]14=5^x[/tex] is equivalent to the given logarithmic equation.
Which is the Exponential function?The exponential function that is represented as y=[tex]m^x[/tex], where:
m=base, m>0
x= exponent
Logarithm PropertiesKnowing some of the main logarithm rules.
Product Rule with the same base: you should repeat the base add the logarithms of the factors.example: [tex]log(a*b)= log a + log b[/tex]
Quotient Rule: you should subtract the logarithm of the numerator with the logarithm of the denominator.example: [tex]log\frac{a}{b} = log a - log b[/tex]
Power Rule: you should multiply the exponent by the logarithm of the base
example: [tex]log\frac{a^b} = b*log a[/tex]
For solving this question, you should apply the logarithm rules and rewrite the function as an exponential equation.
[tex]log_5(x)}-{log_5(25)=7[/tex]
[tex]\frac{log_5(x)}{log_5(25)} =7[/tex]
[tex]{log_5(x)}=7{log_5(25)}[/tex]
[tex]{log_5(x)}=7{log_5(5^2)}[/tex]
[tex]{log_5(x)}=7*2[/tex]
[tex]{log_5(x)}=14[/tex]
[tex]x=5^{14^}[/tex]
Therefore, the exponential function is [tex]14=5^x[/tex]
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Which functions are equivalent to f (x) = RootIndex 4 StartRoot 162 EndRoot Superscript x? Check all that apply.
The functions that are equivalent are
f (x) = 162 Superscript StartFraction x Over 4f (x) = (3 RootIndex 4 StartRoot 2 EndRoot) Superscript xf (x) = left-bracket 3 (2 Superscript one-fourth Baseline) right-bracket Superscript xWhat are roots of a number?
The root of a number is the inverse of the exponents. for instance
a squared has an inverse of square root of a.
mathematically:
a^2 = (a)^1/2 has an inverse of √a
How to find the equivalents of the given dataThe data given:
[tex]\sqrt[4]{162^{x} }[/tex]
Solving the given data for the equivalence
[tex]\sqrt[4]{162^{x} }=162^{x/4}[/tex]
The above proves option A correct
[tex]\sqrt[4]{(3*3*3*3*2)^{x} }[/tex]
[tex]\sqrt[4]{(3^{4} *2)^{x} }[/tex]
[tex](3\sqrt[4]{(2) })^{x}[/tex]
The above solution makes option B correct
[tex](3\sqrt[4]{(2) })^{x}=(3{(2)^{1/4} })^{x}[/tex]
The above proves Option D correct
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complete question
Which functions are equivalent to f (x) = RootIndex 4 StartRoot 162 EndRoot Superscript x? Check all that apply.
f (x) = 162 Superscript StartFraction x Over 4
f (x) = (3 RootIndex 4 StartRoot 2 EndRoot) Superscript x
f (x) = 9 RootIndex 4 StartRoot 2 EndRoot Superscript x
f (x) = 126 Superscript StartFraction 4 Over x
f (x) = left-bracket 3 (2 Superscript one-fourth Baseline) right-bracket Superscript x
Find the slope of the line that passes through (1, 6) and (3, 7)
[tex]m = \frac{y2 - y1}{x2 - x1} \\ m = \frac{7 - 6}{3 - 1} \\ m = \frac{1}{2} [/tex]
I ALSO PROVIDED YOU WITH THE FORMULA OF FINDING THE GRADIENT/SLOPE FOR FUTURE USE
HOPE THIS HELPS.
Answer: Y= 1/2x + 5.5
Function: g(x) = 2x2 – 8
For x ≥ 0, the inverse function is f(x)= √
1
2
x + 4
For x ≤ 0, the inverse function is d(x)= – √
1
2
x + 4
A 3-column table has 3 rows. The first column is labeled x with entries negative 8, 0, 10. The second column is labeled f (x) with entries 0, r, s. The third column is labeled d (x) with entries q, negative 2, t.q =
r =
s =
t =
The inverse function of g(x) = 2x² - 8 for each case is given as follows:
x ≥ 0: y = √(0.5x + 4).x < 0: y = -√(0.5x + 4).Inverse functionTo find the inverse function, we exchange the variables x and y in the original function, and then isolate the variable y.
The original function in this problem is given as follows:
g(x) = 2x² - 8.
The function is not one-to-one, hence the inverse is different for x ≥ 0 and for x < 0.
Exchanging the variables, we have that:
x = 2y² - 8.
Isolating the variable y, we have that:
2y² = x + 8
y² = 0.5x + 4 (the entire equation was divided by 2 to isolate y)
y = ± sqrt(0.5x + 4).
For non-negative values, the inverse function is:
y = sqrt(0.5x + 4) = √(0.5x + 4).
For negative values, the inverse function is:
y = -sqrt(0.5x + 4) = -√(0.5x + 4).
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Answer: 0,2,3,-3
Step-by-step explanation:
The remainder when the expression x^3+6x^2+x+c is divided by x-2 is twice the remainder when the expression is divided by x-1.
show that c=24
Answer:
c = 18
Step-by-step explanation:
According to the remainder theorem, the remainder when f(x) is divided by (x - n) is f(n).
As per question we are given:
f(2) = 2f(1)Substitute and solve for c:
2³ + 6*2² + 2 + c = 2(1³ + 6*1² + 1 + c)8 + 24 + 2 + c = 2(1 + 6 + 1 + c)34 + c = 2(8 + c)34 + c = 16 + 2c2c - c = 34 - 16c = 18Note: I got a different value of c. This may be a result of a typo in the given expression. I provided a guide to solve such problems. Let me know is anything is unclear.
i’ll mark brainliest!
1. (a)The vertex of the function [tex]y=x^{2} +5x-7[/tex] is [tex](-\frac{5}{2} ,-\frac{53}{4} )[/tex]
How is the vertex calculated?
In function [tex]y=x^{2} +5x-7[/tex],
a= 1 ,b= 5, c= -7
For a function [tex]y=ax^{2} +bx+c[/tex] , where (a≠0),
The vertex is [tex](-\frac{b}{2a} ,\frac{4ac-b^{2} }{4a} )[/tex]
[tex]x_v=-\frac{b}{2a}\\\\ =-\frac{5}{2*1} \\\\=-\frac{5}{2} \\\\\\y_v=\frac{4ac-b^{2} }{4a} \\\\=\frac{4*1*(-7)-(5^{2} )}{4*1}\\\\ =-\frac{53}{4}[/tex]
So, vertex [tex](-\frac{5}{2} ,-\frac{53}{4} )[/tex] is minimum because a>0
(b)The solutions are [tex](\frac{\sqrt{53} -5}{2},0) (\frac{-5-\sqrt{53} }{2},0)[/tex]
How is the solution calculated?
When a function [tex]y=ax^{2} +bx+c =0[/tex] where (a≠0),
[tex]x=\frac{-b\pm\sqrt{b^{2} -4ac} }{2a} \\\\x=\frac{-5\pm\sqrt{5^{2} -4(1)(-7)} }{2(1)} \\\\x=\frac{-5\pm\sqrt{53} }{2} \\\\x=\frac{\sqrt{53} -5}{2} , x=\frac{-5-\sqrt{53} }{2} \\\\\text{The solutions are } (\frac{\sqrt{53} -5}{2},0) (\frac{-5-\sqrt{53} }{2},0)[/tex]
2. (a)The vertex of the function [tex]y=-x^{2} +3x+8[/tex] is [tex](\frac{3}{2} ,\frac{41}{4} )[/tex]
How is the vertex calculated?
In function [tex]y=-x^{2} +3x +8[/tex]
a= -1 ,b= 3, c= 8
For a function [tex]y=ax^{2} +bx+c[/tex] , where (a≠0),
The vertex is [tex](-\frac{b}{2a} ,\frac{4ac-b^{2} }{4a} )[/tex]
[tex]x_v=-\frac{b}{2a}\\\\ =-\frac{3}{2*(-1)} \\\\=\frac{3}{2} \\\\\\y_v=\frac{4ac-b^{2} }{4a} \\\\=\frac{4*(-1)*(8)-(3^{2} )}{4*(-1)}\\\\ =\frac{41}{4}[/tex]
So, vertex [tex](\frac{3}{2} ,\frac{41}{4} )[/tex] is maximum because a<0
(b)The solutions are [tex](\frac{3+\sqrt{41} }{2},0) (\frac{3-\sqrt{41} }{2},0)[/tex]
How is the solution calculated?
When a function [tex]y=ax^{2} +bx+c =0[/tex] where (a≠0),
[tex]x=\frac{-b\pm\sqrt{b^{2} -4ac} }{2a} \\\\x=\frac{-3\pm\sqrt{3^{2} -4(-1)(8)} }{2(-1)} \\\\x=\frac{-3\pm\sqrt{41} }{-2} \\\\x=\frac{3\pm\sqrt{41} }{2}\\\\x=\frac{3-\sqrt{41}}{2} , x=\frac{3+\sqrt{41} }{2} \\\\\text{The solutions are } (\frac{3+\sqrt{41} }{2},0) (\frac{3-\sqrt{41} }{2},0)[/tex]
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what’s the answer ???
Answer:
a=29. b=32 c=30. d=30
Step-by-step explanation:
look at the arrow and get the answer if am wrong let me know
3(5y-7)-2(9y-11)=4(8y-13)-17
Answer:[tex]3(5y - 7) - 2(9y - 11) = 4(8y - 13) - 17 \\➪ 15y - 21 - 18y + 22 = 32y - 52 - 17 \\ ➪15y - 18y - 32y = - 52 - 17 + 21 - 22\\➪-35y = - 70 \\➪35y = 70 \\ \: (cancel \: \: out \: "-" \: from \: both \: sides) \\ ➪y = \frac{ 70}{35} = 2 \\ \\ [/tex]
Tʜᴀɴᴋs| (• ◡•)|Hᴏᴘᴇ ɪᴛ ʜᴇʟᴘsa textbook store sold 324 a combined total of chemistry and biology textbooks in a week. the number of chemistry textbooks sold was three times the number of biology textbooks sold. how many textbooks of each type were sold?
Total number of Chemistry textbooks sold is 81 and total number of Biology textbooks sold is 243
How is the number of each textbooks sold, calculated?Let there are x biology textbooks sold.
No of chemistry textbooks sold = 3 the number of biology textbooks sold
ATQ,
A textbook store sold a combined total of 324 biology and chemistry textbooks in a week.
x + 3x = 324
4x = 324
x = 81
It means there are 81 biology textbooks sold and 3x = 3(81) = 243 chemistry textbooks sold.
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0.194805194805...
Convert the decimal into a fraction.
Answer:
The fraction is 15/77Step-by-step explanation:
The repeated part is 194805, six digits. Let the fraction be x.
Convert as follows:
1000000x - x = 194805.194805 ... - 0.194805 ... 999999x = 194805x = 194805/999999Find prime factors of both numerator and denominator and simplify by cancelling common factors:
194805 = 3⁴*5*13*37,999999 = 3³*7*11*13*37.The common factors are:
3³*13*37When they cancel we are left with:
x = (3*5)/(7*11) x = 15/77 Mrs. Smith made 4 pizzas .with 11 oz of cheese How much cheese is
each pizza made with ?
Answer:
2.75
divide 11/4
identify the paragraph proof for the two-column proof. Given: m∠d=125° and m∠r=125° Prove: ∠d≅∠r Two-Column Proof 1. m∠d=125° (Given) 2. m∠r=125° (Given) 3. m∠d=m∠r (Trans. Prop. of = ) 4. ∠d≅∠r (Def. of ≅∠s)
The two column proof is written as follows
Statement Reason
1. m ∠ d = 125° Given
2. m ∠ r = 125° Given
3. m ∠ d = m ∠ r Definition of equality
4. ∠ d ≅ ∠ r Definition of congruent angles
What is two column proof?This is a method of proof used in geometry. It helps to show how the required proof came to be possible. The major items in two column proof is:
StatementReasonwhat is equality?The term equality as used in mathematics is defined to mean that the terms in comparison can be replace by the other
Using the given question as an instance:
m ∠ d = 125° and m ∠ r = 125°
This means that the two angles have same value and they can replace each other.
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There are 3 consecutive integers with a sum of 45. What are the integers?
The three consecutive integers are 14, 15, and 16.
Assume the three consecutive integers to be x-1, x, and x+1.
Sum of all three = 45.
x-1 + x + x+1 = 45
3x = 45 (-1 and +1 get cancelled)
x = 45/3
x = 15
the first integer is x-1 = 15 - 1 = 14
the second integer is x = 15
the third integer is x+1 = 15 + 1 = 16
So, the three consecutive integers are 14, 15, and 16.
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PLS HELP!!
Answers: 8,20,35,12
Answer:
the answer is 8.
Step-by-step explanation:
Use the Pythagoras Theorem since it is a triangle.
[tex]a^{2}+b^{2}= c^{2}\\\\a^{2}+15^{2}= 17^{2}\\ a^{2}+225=289\\ a^{2}=289-225\\ a^{2}= 64\\ \sqrt{a}= \sqrt{64}\\ a=8[/tex]
Since you have the hypotenuse and base, just put those into the equation and find the missing side.
Can someone please help me! I have no idea what I am doing and have no book to look from!
Answer:
Step-by-step explanation:
1) Find DGE
Bisecting an angle makes it half. A straight line is 180
180/2 = DGE
90 = DGE
2) Find FGE
Bisecting an angle makes it half. A straight line is 180
180/2 = FGE
90 = FGE
Denasia and Kenya are training to run a half marathon. In the first week of training, Denasia runs 11 miles and Kenya runs 14 miles.
Each week, Denasia increases her weekly total by 2 miles, and Kenya increases her weekly total by 1.5 miles. Write and solve an
equation to find how many weeks after the first week of training Denasia and Kenya will be running the same number of miles.
A. 2w + 14 = 1.5w+11; w=-6 weeks
B. 2w + 11 = 1.5w+14; w=6 weeks
C. 11w+2= 14w +1.5; w=0, 167 weeks
Option B. 2w + 11 = 1.5w + 14, w = 6 weeks is the correct answer
Miles run by Denasia = 11 miles
Miles run by Kenya = 14 miles
Let w represent the number of weeks after which both Kenya and Denasia will be running the same number of miles
Formulating the equation we get:
Miles run by Denasia + Increase in miles each week*Number of weeks = Miles run by Kenya + Increase in miles each week*Number of weeks
11 + 2w = 14 + 1.5w
0.5w = 3
w = 6
Hence, option B is the correct answer according to the given contraints and both Denasia and Kenya will run the same distance in the 6th week
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n/−3 +5>4 Solve for this
Answer:
N<3
Step-by-step explanation:
9(w + 1) = 99
w =
if you tell me the answer then ur a W
w = 10.
Step-by-step explanation:
9(w+1) = 99
9w+9 = 99
9w = 99-9
9w = 90
w = 90/9
w = 10.
[tex]{ \boxed{ \red{ \sf{w = 10}}}}[/tex]
Step-by-step explanation:
[tex]{ \purple{ \sf{9(w + 1) = 99}}}[/tex]
[tex]{ \purple{ \sf{9w + 9 = 99}}}[/tex]
[tex]{ \purple{ \sf{9w = 99 - 9}}}[/tex]
[tex]{ \purple{ \sf{9w = 90}}}[/tex]
Divide both the sides by 9 then
[tex]{ \purple{ \sf{ \frac{9w}{9} = \frac{90}{9} }}}[/tex]
[tex]{ \pink{ \boxed{ \blue{ \sf{w = 10}}}}}[/tex]
3. a person is watching the space shuttle launch. the person is 3000 ft from the launch pad. how fast is the distance between the person and the shuttle changing when the shuttle is 4,000 ft high and rising at a rate of 800 ft/sec?
The distance between the person and the shuttle is changing at a rate of 640 feet/sec when the shuttle is 4000 ft high and is rising at a rate of 800 ft/sec.
Considering x to be the height of the triangle and y to be the hypotenuse of the triangle, a right-angled triangle will be formed according to the given information having a base of 3000 feet which is the distance of the person from the launch pad.
In this right-angle triangle x will be the height of the shuttle and y will be the distance between the person and the shuttle.
Now we apply the Pythagorean theorem to the triangle;
y² = x²+(3000)²
Now differentiating this equation with respect to time,t :
2y (dy/dt) = 2x(dx/dt) + 0
dy/dt = x(dx/dt)/y
As the shuttle is rising at a speed of 800 ft/sec, (dx/dt)=800
Substitute 4000 for x into the equation, y²=x²+(3000)², to find y;
y² = (4000)²+(3000)²
y = 5000
Now we substitute 4000 for x, 5000 for y, and 800 for dx/dt in the equation, dy/dt = x(dx/dt)/y
dy/dt = 4000(800) / 5000
dy/dt = 640
Therefore, the distance between the person and the shuttle is increasing at a rate of 640 feet/sec.
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Compare these two fractions: 3/10 and 4/5*03/10 > 4154/5>3/10О4/5 = 3/10
suppose you give 10 people a taste test where they each try samples of two different brands of soda. after drinking both samples they tell you which soda they liked best. what is a/the confounding variable in this experiment?
Confounding variables are any other variable that also has an effect on your dependent variable.
What is a confounding variable?
A confounding variable, also called a confounder or confounding factor, is a third variable in a study examining a potential cause-and-effect relationship.
A confounding variable is related to both the supposed cause and the supposed effect of the study. It can be difficult to separate the true effect of the independent variable from the effect of the confounding variable.
In your research design, it's important to identify potential confounding variables and plan how you will reduce their impact.
For example, if you are researching whether drinking both sample lead to taste test, then drinking both sample is your independent variable and taste test is your dependent variable. Confounding variables are any other variable that also has an effect on your dependent variable.
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Simplify 5(−4−8w)−3w.
Answer:
-20-43w
Step-by-step explanation:
5(-4-8w)-3w
-20-40w-3w
-20-43w
how many of these figures have atleast one vertex
HELP MEEE HELP ME HLSP ME
The inequality for the temperature in the house measured by thermostat is; 24° ≤ t ≤ 26°. The highest temperature is 26° C.
What is defined as the term inequality?In mathematics, an inequality is a relationship representations or values that aren't equal to each other. When two numbers are equal, we use the symbol '=', and if they are not equal, we use the symbol ≠.For the given question.
The thermostat at the house is set at 25° C.
Now, there is a deviation of the temperature of 1° C.
Thus,
The lowest temperature measured = 25° C - 1° C = 24° C.
The highest temperature measured = 25° C + 1° C = 26° C.
Let 't' be the temperature of the house.
The inequality will be;
24° ≤ t ≤ 26°
Thus, the maximum value of the temperature will be 26° C.
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Which number is the smallest?
pls answer fast
Responses
A 4.68 x 10−44.68 x , 10 -4
B 5.48 x 10−85.48 x , 10 -8
C 2.85 x 10−62.85 x , 10 -6,
D 3.24 x 10−53.24 x , 10 -5,
E 1.28 x 10−4
The smallest number from the given options is
[tex]5.48 × {10}^{ - 8} [/tex]
The correct answer option is option d
How to find the smallest number?A.
[tex]1.28 × {10}^{ - 4} [/tex]
= 1.28 × 0.0001
= 0.000128
B.
[tex]2.85 × {10}^{ - 6} [/tex]
= 2.85 × 0.000001
= 0.00000285
C.
[tex]3.24 × {10}^{ - 5} [/tex]
= 3.24 × 0.00001
= 0.0000324
D.
[tex]5.48 × {10}^{ - 8} [/tex]
= 5 48 × 0.00000001
= 0.0000000548
Therefore, the smallest number is
[tex]5.48 × {10}^{ - 8} [/tex]
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y > x-2 3 4 y> -x + 1 Which point is a solution set to the inequalities? b (0,1) (1.0) (-3, 2) (2.-3) d
Replace each point in both inequalities and see if the inequality remains:
(0,1) = (x,y)
1 ≥ 1/3(0)-2
1≥-2
y>4/3x+1
1>4/3(0)+1
1>1 NOT TRUE.
Point b.
(1,0) = (x,y)
0 ≥ 1/3(1)-2
0≥ 1/3-2
0≥-5/3
0>4/3(1)+1
0>7/3 NOT TRUE
Pint c. (-3,2)
2≥1/3(-3)-2
2≥ -1-2
2≥-3
2>(4/3)-3+1
2>-4+1
2>-3 TRUE
(-3,2) is a solution point to the set of inequalities. (option c)
Which of the contexts below could not be modeled by an exponential function?
A certain population of 16 aggressive zombies quintuples every day.
A taxi charges a flat fee of $2.00 for pick-up, then an additional fee of 4.75 per mile.
A coin collection appreciates at a rate of 5.2% per year.
A car depreciates at a rate of 4.9% per year.
The answer choice which could not be modelled by an exponential function among the given answer choices is; A taxi charges a flat fee of $2.00 for pick-up, then an additional fee of 4.75 per mile.
Which context could not be modelled by an exponential function?It follows from the task content that the expression which could not be modelled by an exponential expression be identified among the answer choices.
An exponential function is one in which case the change factor is expressed as an exponent of the independent variable.
However, the function described as; A taxi charges a flat fee of $2.00 for pick-up, then an additional fee of 4.75 per mile is termed a linear model in which case;
The rate of change is constant and is equal to; $4.75 per mile.
Hence the required answer choice which cannot be modelled by an exponential function is; A taxi charges a flat fee of $2.00 for pick-up, then an additional fee of 4.75 per mile.
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The figure below shows a triangle with vertices A and B on a circle and vertex C outside it. Side AC is tangent to the circle. Side BC is a secant intersecting the circle at point X.What is the measure of angle ACB? (6 points)O 29°O 8°O 16°O 21°
The Solution.
Considering triangle ABC, we have
[tex]\begin{gathered} angle\text{ACB}=180-(90+40+42) \\ =180-172=8^o \end{gathered}[/tex]Hence, the correct answer is 8 degrees (option B)
The next two problems show the attempts of two different students to provide a proof of the same statement. Each proof has something about it that is not quite correct. Explain how you would improve the argument.
We will improve the argument by stating that two intersecting lines have vertically opposite angles equal in magnitude.
We are given a diagram in which we have two lines, namely "m" and "n".The intersection point of the two lines is P.We need to prove that ∠1≅∠3.We know that line "m" is a straight line, so it makes a total angle of 180° on either of its sides.∠1+∠2 = 180°We know that line "n" is a straight line, so it makes a total angle of 180° on either of its sides.∠3+∠2 = 180°From the above two equations, we find that :∠1+∠2 = 180° = ∠3+∠2∠1+∠2 = ∠3+∠2∠1 = ∠3Thus, ∠1≅∠3.To learn more about lines, visit :
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