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3 years ago

Tionna has saved dimes and quarters in her piggy bank. Define the variables

Mathematics

1 answer:

olya-2409 [2.1K]3 years ago

4 0

Step-by-step explanation:

y = .10d + .25q

y is the total amount, if you have d dimes and q quarters

You might be interested in

Evaluate the integral, show all steps please!

Aloiza [94]

Answer:

$\displaystyle \int \dfrac{1}{(9-x^2)^{\frac{3}{2}}}\:\:\text{d}x=\dfrac{x}{9\sqrt{9-x^2}} +\text{C}$

Step-by-step explanation:

Fundamental Theorem of Calculus

$\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))$

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

$\displaystyle \int \dfrac{1}{(9-x^2)^{\frac{3}{2}}}\:\:\text{d}x$

Rewrite 9 as 3² and rewrite the 3/2 exponent as square root to the power of 3:

$\implies \displaystyle \int \dfrac{1}{\left(\sqrt{3^2-x^2}\right)^3}\:\:\text{d}x$

Integration by substitution

$\boxed{\textsf{For }\sqrt{a^2-x^2} \textsf{ use the substitution }x=a \sin \theta}$

$\textsf{Let }x=3 \sin \theta$

$\begin{aligned}\implies \sqrt{3^2-x^2} & =\sqrt{3^2-(3 \sin \theta)^2}\\ & = \sqrt{9-9 \sin^2 \theta}\\ & = \sqrt{9(1-\sin^2 \theta)}\\ & = \sqrt{9 \cos^2 \theta}\\ & = 3 \cos \theta\end{aligned}$

Find the derivative of x and rewrite it so that dx is on its own:

$\implies \dfrac{\text{d}x}{\text{d}\theta}=3 \cos \theta$

$\implies \text{d}x=3 \cos \theta\:\:\text{d}\theta$

Substitute everything into the original integral:

$\begin{aligned}\displaystyle \int \dfrac{1}{(9-x^2)^{\frac{3}{2}}}\:\:\text{d}x & = \int \dfrac{1}{\left(\sqrt{3^2-x^2}\right)^3}\:\:\text{d}x\\\\& = \int \dfrac{1}{\left(3 \cos \theta\right)^3}\:\:3 \cos \theta\:\:\text{d}\theta \\\\ & = \int \dfrac{1}{\left(3 \cos \theta\right)^2}\:\:\text{d}\theta \\\\ & = \int \dfrac{1}{9 \cos^2 \theta} \:\: \text{d}\theta\end{aligned}$

Take out the constant:

$\implies \displaystyle \dfrac{1}{9} \int \dfrac{1}{\cos^2 \theta}\:\:\text{d}\theta$

$\textsf{Use the trigonometric identity}: \quad\sec^2 \theta=\dfrac{1}{\cos^2 \theta}$

$\implies \displaystyle \dfrac{1}{9} \int \sec^2 \theta\:\:\text{d}\theta$

$\boxed{\begin{minipage}{5 cm}\underline{Integrating $\sec^2 kx$}\\\\$\displaystyle \int \sec^2 kx\:\text{d}x=\dfrac{1}{k} \tan kx\:\:(+\text{C})$\end{minipage}}$

$\implies \displaystyle \dfrac{1}{9} \int \sec^2 \theta\:\:\text{d}\theta = \dfrac{1}{9} \tan \theta+\text{C}$

$\textsf{Use the trigonometric identity}: \quad \tan \theta=\dfrac{\sin \theta}{\cos \theta}$

$\implies \dfrac{\sin \theta}{9 \cos \theta} +\text{C}$

$\textsf{Substitute back in } \sin \theta=\dfrac{x}{3}:$

$\implies \dfrac{x}{9(3 \cos \theta)} +\text{C}$

$\textsf{Substitute back in }3 \cos \theta=\sqrt{9-x^2}:$

$\implies \dfrac{x}{9\sqrt{9-x^2}} +\text{C}$

Learn more about integration by substitution here:

brainly.com/question/28156101

brainly.com/question/28155016

4 0

2 years ago

An artis creates an abstract sculpture that appears as a box perched on top of a post the box is in the shape of a cube with edg

s344n2d4d5 [400]

Volume of the box = length ^ 3 = 5^3 = 125 cubic inches
Volume of rectangular prism = length x width x height = 3 x 4 x 18 = 216 cubic inches
Volume of entire sculpture = volume of box + volume of rectangular prism = 125 + 216 = 341 cubic inches.

4 0

3 years ago

9x – 1= 10x + 9 What is x equal to

ivolga24 [154]

Answer:

-10=x

Step-by-step explanation:

9x-1=10x+9

9x-10×=9+1

-x=10

x=-10

4 0

3 years ago

Shaun read 60 pages in 3/4 h. What is the unit rate that represents the number of pages Shaun can read in 1 h?

Alexus [3.1K]

Answer:

60=3/4

x=4/4

4/4÷1/4

4/4*4/1

4/1*1/1

4/1

3/4=12 because if 1/4 is 4 then 3/4 is 12

60=12

60÷12=5

5*4/20

60+20=80

Shaun will read 80 pages in 1 hour

I can only read 25 because i hate reading

4 0

3 years ago

Admission to a baseball game is $3.00 for general admission and $5.50 for reserved seats. The receipts were $3277.50 for 845 pai

a_sh-v [17]

Answer:

548, 297

Step-by-step explanation:

We can set this up as a system of equations. The first equation is for how many tickets were sold, which is x+y=845. The second equation is for the cost of the tickets, which is 3x+5.5y=3277.50. The best method for this problem (other than graphing) is elimination. First, we need to multiply one equation to get the opposite of one of the variables. The easiest one to do for this problem is x. The opposite of 3 is -3, so we have to multiply ALL of the first equation by -3. If you do this, then you get -3x-3y=-2535. Now, combine the equations:

3x + 5.5y = 3277.50

-3x -3y = -2535

________________

0x+2.5y=742.5

Now, isolate y by dividing both sides by 2.5. If you do this, you get y=297. Now, plug this in for the y in the first equation. Isolate x.

x+297=845

-297 -297

__________

0 548

x=548

Hope this helps!

7 0

2 years ago