A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to r(t)=15√t+2, find the area of the ripple as a function of time. Find the area of the ripple at t=2 .
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FiThe radius, in inches, grows as a function of time in minutes according to:
[tex]r(t)=15\sqrt{t+2}[/tex]We know that the area of a circle is given by:
[tex]A=\pi r^2[/tex]Where r is the radius of the circle. Then, using r(t) in this equation:
[tex]\begin{gathered} A(t)=\pi\cdot\lbrack r(t)\rbrack^2=\pi\lbrack15\sqrt{t+2}\rbrack^2 \\ \\ \therefore A(t)=225\pi(t+2) \end{gathered}[/tex]Finally, we evaluate this function for t = 2:
[tex]\begin{gathered} A(2)=225\pi(2+2)=225\pi(4) \\ \\ \therefore A(2)=900\pi\text{ in}^2 \end{gathered}[/tex]Related Questions
Differentiate a trig function that is greater than a power of 1, and involve either quotient, chain, or product rule.Differentiate a sine and cosine function that involves product and chain rule. Find the equation of the tangent line at x = a special triangle point (i.e. /4, /6, /3).Differentiate a function that involves both trig and exponential functions.[hint: add your own twist to this question for level 3/4]Differentiate an exponential function. [hint: add your own twist to this question for level 3/4]Differentiate a function where you have “y” and “x” on both sides of the equation and they cannot be simplified by collecting like terms or isolating y (i.e. y on one side and y^2 on the other). [hint: add your own twist to this question for level 3/4]
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Solution:
Given a trigonometric function that is greater than power of 1 as shown below:
[tex]y=sin^2x\text{ ---- equation 1}[/tex]To differentiate the function, we use the chain rule.
According to the chain rule,
[tex]\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}[/tex]From equation 1, let
[tex]u=sin\text{ x --- equation 2}[/tex]This implies that
[tex]\begin{gathered} y=u^2 \\ \Rightarrow\frac{dy}{du}=2u \end{gathered}[/tex]From equation 2,
[tex]\begin{gathered} \begin{equation*} u=sin\text{ x} \end{equation*} \\ \Rightarrow\frac{du}{dx}=cos\text{ x} \end{gathered}[/tex][tex]\begin{gathered} Recall\text{ from the chain rule:} \\ \frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx} \\ \Rightarrow2u\times\text{cos x} \\ \frac{dy}{dx}=2ucos\text{ x} \\ but\text{ } \\ u=sin\text{ x} \\ \therefore\frac{dy}{dx}=2(sin\text{ }x)(cos\text{ }x) \end{gathered}[/tex]What is the range 12 ,20,18,25,6
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The maximum of data is 25
The minimum of data is 6
Then, the range is:
range = maximum - minimum
range = 25 - 6
range = 19
Could you help me with how to multiply polynomials(5x - 1)(2x^2 -3x + 4)
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ANSWER:
[tex](5x\: -\: 1)(2x^2\: -3x\: +\: 4)=10x^3-17x^2+23x-4[/tex]STEP-BY-STEP EXPLANATION:
We have the following multiplication of polynomials:
[tex]\mleft(5x-1\mright)\mleft(2x^2-3x+4\mright)[/tex]When multiplying two polynomials we must bear in mind that all the terms of the first polynomial must be multiplied by all the terms of the second polynomial, like this:
[tex]\begin{gathered} \mleft(5x-1\mright)\mleft(2x^2-3x+4\mright)=5x\cdot2x^2+5x\cdot-3x+5x\cdot4+(-1)\cdot2x^2+(-1)\cdot-3x+(-1)\cdot4 \\ (5x\: -\: 1)(2x^2\: -3x\: +\: 4)=10x^3-15x^2+20x-2x^2+3x-4 \\ (5x\: -\: 1)(2x^2\: -3x\: +\: 4)=10x^3-17x^2+23x-4 \end{gathered}[/tex]Is there any other further step I need to do? The answer is very close but not exact so I’m unsure.
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Given the matrices:
[tex]A=\begin{bmatrix}{10} & {4} & {0} \\ {1} & {3} & {1}\end{bmatrix},B=\begin{bmatrix}{4} & {1} & \\ {2} & {2} & {} \\ {0} & {-1} & \end{bmatrix}[/tex]we will find the value of AB + I
First, we will find the product of AB as follows:
[tex]AB=\begin{bmatrix}{10} & {4} & {0} \\ {1} & {3} & {1}\end{bmatrix}\cdot\begin{bmatrix}{4} & {1} & \\ {2} & {2} & {} \\ {0} & {-1} & \end{bmatrix}=\begin{bmatrix}{10\cdot4+2\cdot4+0\cdot0} & {1\cdot01+4\cdot2+0\cdot-1} & {} \\ {1\cdot4+3\cdot2+1\cdot0} & {1\cdot1+3\cdot2+1\cdot-1} & {} \\ {} & {} & {}\end{bmatrix}[/tex]simplifying the answer:
[tex]AB=\begin{bmatrix}{48} & {18} & {} \\ {10} & {6} & {} \\ {} & {} & {}\end{bmatrix}[/tex]Now, we will add the unity matrix to the answer:
[tex]AB+I=\begin{bmatrix}{48} & {18} & {} \\ {10} & {6} & {} \\ {} & {} & {}\end{bmatrix}+\begin{bmatrix}{1} & {0} & {} \\ {0} & {1} & {} \\ {} & {} & {}\end{bmatrix}=\begin{bmatrix}{49} & {18} & {} \\ {10} & {7} & {} \\ {} & {} & {}\end{bmatrix}[/tex]So, the answer will be option D
1. What is the other endpoint of the segment with midpoint -3 and endpoint -7? A-11 01 D 4 B -5 2. The endngints of ST are S(2,-2) and T14, 2). What are the coordinates of the
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We have a segment with points S and T.
We know the coordinates of S=(2,-2) and the midpoint M=(-
Can someone please help me find the value of X?
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Remember that
the sum of the interior angles in any polygon is equal to
S=180(n-2)
where
n is the number of sides of polygon
In this problem
we have
n=6 (hexagon)
so
substitute
S=180(6-2)
S=720 degrees
step 2
Adds the interior angles
720=120+(5x-6)+(4x+14)+(7x)+(8x-8)+(6x)
solve for x
combine like terms
720=30x+120
30x=720-120
30x=600
x=20Write a quadratic equation with 7 and 2/5 as its roots. Write the equation in the form ax2 + bx+c= 0, where a, b, and c are integers.
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As given by the question
There are given that the roots: 7 and 2/5.
Now,
Since the roots are integers, we can write the equation in the given form using a = 1.
Then,
b is the opposite of the sum of the roots
So,
[tex]\begin{gathered} b=-((7)+(\frac{2}{5})) \\ b=-(\frac{35+2}{5}) \\ b=-\frac{37}{5} \end{gathered}[/tex]And
c is the products of the roots
So,
[tex]\begin{gathered} c=7\times\frac{2}{5} \\ c=\frac{14}{5} \end{gathered}[/tex]Now,
The desired quadratic equation is:
[tex]\begin{gathered} ax^2+bx+c=0 \\ x^2-\frac{37}{5}x+\frac{14}{5}=0 \\ 5x^2-37x+14=0 \end{gathered}[/tex]Hence, the correct option is A.
The table below shows the cost of downloading songs from a website.Number of Songs Total Cost11$10.5613$12.4818.$17.28At this rate, what is the cost per song?Answer: $per song
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To know the cost per song we make a division between the total Cost and the number of songs, then we can take any pair of data
I will use 11 songs and $10.56
[tex]\frac{10.56}{11}=\frac{24}{25}=0.96[/tex]to check we can use another pair (18 songs and $17.28)
[tex]\frac{17.28}{11}=\frac{24}{25}=0.96[/tex]then the cost per song is $0.96
Amelia bought spider rings for Halloween goodie bags. She bought 13 packs of red rings, 16 packs of yellow rings, and 14 packs of green rings. If each pack had 12 rings, how many rings did Amelia buy?
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We know that
• She bought 13 packs of red rings.
,• She bought 16 packs of yellow rings.
,• She bought 14 packs of green rings.
,• Each pack has 12 rings.
This problem is about multiplication, notice that each pack includes 12 rings, that means we need to multiply each pack by 12, in order to find the total number of rings of each color.
[tex]R=13\cdot12=156[/tex]There are 156 red rings.
[tex]Y=16\cdot12=192[/tex]There are 192 yellow rings.
[tex]G=14\cdot12=168[/tex]There are 168 green rings.
Now, we sum all these numbers to find the total
[tex]T=168+192+156=516[/tex]Therefore, there are 516 rings in total.Translate to a system Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
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Given:
Let x be the number of the cards and y be the number of the package.
Given that the number of cards is at least 4 more than twice the number of packages.
[tex]x=2y+4[/tex]Given that mailing costs no more than $500 and the cost of mailing a card is $3 and for a package, the cost is $7.
[tex]3x+7y=500[/tex]Substitute x=2y+4 in this equation, we get
[tex]3(2y+4)+7y=500[/tex][tex]6y+8+7y=500[/tex][tex]13y=500-8[/tex][tex]y=\frac{492}{13}[/tex][tex]y=37.8[/tex]Let y=37 and substitute in x=2y+4, we get
[tex]x=2\times37+4[/tex][tex]x=78[/tex]Hence the number of cards = 78 and the number of packages =37.
The total cost for this is $493 not more than $500.
the sum of 6 times a number and 8 equals 7? translate into equation
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Let:
x = Unknown number
the sum of 6 times a number:
[tex]6x+[/tex]and 8 equals 7, so:
[tex]\begin{gathered} 6x+8=7 \\ \text{solve for x:} \\ \text{subtract 8 from both sides:} \\ 6x=7-8 \\ 6x=-1 \\ \text{divide both sides by 6:} \\ x=-\frac{1}{6} \end{gathered}[/tex]Type the correct answer in the box. Use numericals instead of words. 5 less than a number is equivalent to 1 more than three times the number. The number is _____.
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Answer:
2
Explanation:
Let the number be x
5 less than a number is expressed as x - 5
1 more than three times the number is expressed as 3x + 1
Equate both expression and find the number
x - 5 = 3x+1
x - 3x = 1 - 5
-2x = -4
x = -4/-2
x = 2
Hence the number is 2
Thomas is married and files jointly with his spouse. Their combined taxable income is $25,799. Their employers withheld $4,386 in taxesfor the year. Determine theamount to be refundedor the balance due.Circle one: RefundBalance Due
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EXPLANATION
As we can see on the table, the amount to be refunded is equivalent to the difference between $3,866 and $4,386, so it is $520
How many pounds of candy that sells for $0.85 per Ib must be mixed with candy that sells for $1.22 per lb to obtain 9 lb of a mixture that should sell for $0.92 per lb? 50.85-per-lb candy: 73 lb (Type an integer or decimal rounded to two decimal places as needed.) $1.22-per-b candy
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This system gives two equations
[tex]\frac{0.85x+1.22y}{9}=0.92[/tex][tex]x+y=9[/tex]where x is the number pounds of $0.85/lb candy and y is the number of pounds of $1.22/lb candy.
The solution to the system is
[tex]x=7.297[/tex][tex]y=1.70[/tex]Hence, 7.297 lb of $0.85 candy is required in order that if we mix them with 1.70 lb of $1.22 candy, we will get a 9 lb solution of 0.92 /lb candy.
Which point shows the number with the greatest absolute value? А B + + D TH 30 40 50 -50 -40 -30 - 20 -10 0 10 20 O Point A O Point B O Point C O Point D ہ
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We are shown points A, B, C, and D on a number line.
We are asked to find out which point shows the number with the greatest absolute value?
Recall that the absolute value of a number is always positive.
The negative values of points A and B will become positive.
As you can see from the number line, point A is closer to -40 and the point D is closer to 30
The absolute value of point A will be closer to |-40| = 40
Since 40 is greater than 30, point A shows the number with the greatest absolute value.
Therefore, the correct answer is Point A.
1/9=_/54What is the answer?
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can someone please help?just in case if the picture seems blurry, the question says the take off ramp is parallel to the waiting ramp, and the interest ramps are parallel. Given that the measure of angle a is 88 find the measure of each remaining angles
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Vanessa collected Barbie dolls. She began with 2 dolls and added the same amount of dolls to her collection each year. In the 24th year, Vanessa had 98 dolls. Which function, d(n), can be used to determine the number of dolls Vanessa had in any year?
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The correct answer is d(n) = 4n +2
3. A toy box is 24 cm long, 15 cm wide and 11 cm high. What is the volume of the toy box? What is the correct number sentence for this problem? A.V=24×15×11B.V=24×15C.V=24×11D.V=15×11
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ANSWER
[tex]\begin{gathered} V=24*15*11 \\ V=3960\text{ }cm^3 \end{gathered}[/tex]EXPLANATION
The box is a rectangular prism. The volume of a rectangular prism is given by:
[tex]V=L*W*H[/tex]where L = length
W = width
H = height
Therefore, the volume of the box can be written in the number sentence:
[tex]V=24*15*11[/tex]and the volume of the box is:
[tex]V=3960\text{ }cm^3[/tex]That is the answer.
determine whether the binomial expression is a factor to the following polynomial.[tex]p(x) = {x}^{3} - 9x + 1 \: \: \: \: \: \: \: \: \: (x - 3)[/tex]the binomial expression is (x-3) ^^answer choicesA. yesB. no
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We can find if (x-3) is a factor by dividing P(x) by (x-3).
A simpler way is replacing x with 3 and if the value of P(x) is 0, then (x-3) is a factor of P(x). This is because x=3 is a root of P(x) and therefore it can be factorized with the term (x-3).
Then, we calculate P(3):
[tex]P(3)=3^2-9\cdot3+1=9-27+1=-17[/tex]As x=3 is not a root of P(x), then (x-3) is not a factor of P(x).
Answer: No.
which value must be added to the expression x^2 + x to make it a perfect-square trinomial
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A perfect square trinomial is written in the form
[tex]undefined[/tex]Callie's grandmother pledged R150, 00 for every mile Callie walked in her walk-a-thon. Callie walked 14.5 km. How much does her grandmother owe? ( assume 8 km = 5miles)
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Given:-
Callie walked 14.5 km. And also given 8 km=5 miles.
At first we convert 14.5 km to miles. we get,
[tex]\begin{gathered} 14.5\times\frac{5}{8}=\frac{72.5}{8} \\ \text{ =9.0625} \end{gathered}[/tex]So 14.5 km is 9.0625 miles.
Callie's grandmother pledged Rs. 150 for every mile. so for 9.0625 miles it is,
[tex]9.0625\times150=1359.375[/tex]So her grandmother owe Rs. 1359.375
Hi, what is the LCM of the numbers 3 and 15
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Answer:
15
Step-by-step Explanation:
LCM is the least common multiple, that is, is the least number that is a multiple of both numbers.
To find it, first, let's write the multiples of 3:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Now, let's write the multiples of 15:
Multiples of 15: 15, 30, 45, ...
If you compare the multiples of 3 and 5, we can see that they have some common multiples, as 15 and 30.
From this common multiples, 15 is the smallest number. So, 15 is the LCM of 3 and 15.
Answer: 15.
5.Line AB is 14 inches long. What is the approximate area of this circle?АBa. 42 square inchesb. 615 square inchesc. 160 square inchesd. 154 square inches
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The area of a circle is given as
A =
in the diagram, ab to ec are perpendicular. if m
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Write an expression for the operation described.
"5 divided by the product of 3 and 2"
A (5 ÷ 3) × 2(5 ÷ 3) × 2
B 3 × (2 ÷ 5)3 × (2 ÷ 5)
C (3 × 2) ÷ 5(3 × 2) ÷ 5
D 5 ÷ (3 × 2)
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D] 5 ÷ (3 × 2) is the expression for the operation "5 divided by the product of 3 and 2".
The operation "5 divided by the product of 3 and 2" means that number 5 divided by the product of 3 and 2.
The mathematical representation of this operation is 5 ÷ (3 × 2).
The answer to this operation = 5 / 6.
Hence, 5 ÷ (3 × 2) is the expression for the operation "5 divided by the product of 3 and 2".
To understand more about multiplication and division refer -
https://brainly.com/question/28768606
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Moshde runs a hairstyling business from her house. She charges $42 for a haircut and style. Her monthly expenses are $1070. She wants to be able to put at least $1,249 per month into her savings account order to open her own salon. How many "cut & styles" must she do to save at least $1,249 per month?
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ANSWER:
56 cut & styles
STEP-BY-STEP EXPLANATION:
They tell us that each haircut and style charges $42 and that the monthly expenses are $1070, he wants to save a total of $1249, with this information we can establish the following equation:
[tex]42x-1070=1249[/tex]Where x would be the amount of cut & styles, we solve for x:
[tex]\begin{gathered} 42x=1249+1070 \\ x=\frac{2319}{42} \\ x=55.2 \\ x\cong56 \end{gathered}[/tex]If you make a total of 55 cut & styles, the amount does not reach a total of $1249 per month, therefore, at least 56 cut & styles are needed, to achieve the monthly goal.
Prison Sentences The average prison sentence for a person convicted of second-degree murder is 15 years. If the sentences are normally distributed with astandard deviation of 2.1 years, find the following probabilities.P (x> 18) =
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Givens.
• The mean is 15 years.
,• The standard deviation is 2.1 years.
,• x = 18.
Using a graphic calculator, the probability P(X > 18) is 0.0766.
Therefore, the answer is 0.07
Find the height of the cliff. If necessary, round to the nearest hundredth yard.
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We are given a diagram showing a slope, and a vertical height. We now have what represents a right angled triangle. The distance from the base of the cliff to the end of the slope is given as 24 yards. The slope itself is 37 yards. We shall now determine the height of the cliff (from ground to top) as indicated.
Note that we shall use the Pythagoras' theorem which is;
[tex]c^2=a^2+b^2[/tex]Where we have
[tex]\begin{gathered} c=\text{hypotenuse (longest side)} \\ a,b=\text{other sides} \end{gathered}[/tex]We can now substitute the given values/side lengths and we'll have;
[tex]37^2=24^2+b^2[/tex][tex]1369=576+b^2[/tex]Subtract 576 from both sides;
[tex]793=b^2[/tex]Take the square root of both sides;
[tex]\begin{gathered} \sqrt[]{793}=\sqrt[]{b^2} \\ 28.160255\ldots=b \end{gathered}[/tex]Rounded to the nearest hundredth, the answer now becomes;
ANSWER:
[tex]b=28.16yd[/tex]The last option is the correct answer
3x and 8x are like terms.true or false
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Like terms are those terms whose variable and its corresponding exponent are the same. Here we have 3x and 8x. Both terms have the number:
[tex]x^1[/tex]Which means that they have the same variable and the same exponents. Then they are like terms and the answer is True.