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2 years ago

How many 3 1/4 inch pieces of ribbion can be cut from a spool containing 19 1/2 inches of ribbon

Mathematics

1 answer:

goldenfox [79]2 years ago

5 0

6. 6 3 1/4 inch pieces of ribbon can be cut from 19 1/2 inches of ribbon

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Evaluate the interval (Calculus 2)

Darya [45]

Answer:

$2 \tan (6x)+2 \sec (6x)+\text{C}$

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:

$\displaystyle \int \dfrac{12}{1-\sin (6x)}\:\:\text{d}x$

$\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int a\:\text{f}(x)\:\text{d}x=a \int \text{f}(x) \:\text{d}x$\end{minipage}}$

If the terms are multiplied by constants, take them outside the integral:

$\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)}\:\:\text{d}x$

Multiply by the conjugate of 1 - sin(6x) :

$\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)} \cdot \dfrac{1+\sin(6x)}{1+\sin(6x)}\:\:\text{d}x$

$\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{1-\sin^2(6x)} \:\:\text{d}x$

$\textsf{Use the identity} \quad \sin^2 x+ \cos^2 x=1:$

$\implies \sin^2 (6x) + \cos^2 (6x)=1$

$\implies \cos^2 (6x)=1- \sin^2 (6x)$

$\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{\cos^2(6x)} \:\:\text{d}x$

Expand:

$\implies 12\displaystyle \int \dfrac{1}{\cos^2(6x)}+\dfrac{\sin(6x)}{\cos^2(6x)} \:\:\text{d}x$

$\textsf{Use the identities }\:\: \sec \theta=\dfrac{1}{\cos \theta} \textsf{ and } \tan\theta=\dfrac{\sin \theta}{\cos \theta}:$

$\implies 12\displaystyle \int \sec^2(6x)+\dfrac{\tan(6x)}{\cos(6x)} \:\:\text{d}x$

$\implies 12\displaystyle \int \sec^2(6x)+\tan(6x)\sec(6x) \:\:\text{d}x$

$\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})$\end{minipage}}$

$\boxed{\begin{minipage}{6 cm}\underline{Integrating $ \sec kx \tan kx$}\\\\$\displaystyle \int \sec kx \tan kx\:\text{d}x= \dfrac{1}{k}\sec kx\:\:(+\text{C})$\end{minipage}}$

$\implies 12 \left[\dfrac{1}{6} \tan (6x)+\dfrac{1}{6} \sec (6x) \right]+\text{C}$

Simplify:

$\implies \dfrac{12}{6} \tan (6x)+\dfrac{12}{6} \sec (6x)+\text{C}$

$\implies 2 \tan (6x)+2 \sec (6x)+\text{C}$

Learn more about indefinite integration here:

brainly.com/question/27805589

brainly.com/question/28155016

3 0

1 year ago

Solve the system using the substituion method -4x+y=8 x-3y=9

lions [1.4K]

Hi there!

x - 3y = 9
x = 9 + 3y

-4x + y = 8
-4(9+3y) + y = 8
-36 - 12y + y = 8
-36 - 11y = 8
-11y = 44
-y = 4
y = -4

x - 3y = 9
x - 3(-4) = 9
x + 12 = 9
x = 21

So, your answers are y = -4 and x = 21.

Hope this helps!

3 0

3 years ago

Do any of you know how to do this

Gre4nikov [31]

Answer:

$(x + 5)$

Step-by-step explanation:

${x}^{2} + 8x + 15 \\ {x}^{2} + 3x + 5x + 15 \\ x(x + 3)5(x + 3) \\ (x + 3)(x + 5) \\ length \: = \frac{(x + 3)(x + 5)}{(x + 3)} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (x + 5)$

5 0

2 years ago

Read 2 more answers

Five hamburgers and one order of fries together cost $10.24, whereas one hamburger and five orders of fries together cost $5.84.

lubasha [3.4K]

Hamburgers = h
fries = f
5h + f = $10.24
h + 5f = $5.84
6h + 6f = $ 16.08
3h + 3f = $8.04

The correct answer is C.

8 0

3 years ago

I need help on this

Dvinal [7]

Answer:

1/12, 1/10

Step-by-step explanation:

P(Bead bracelet and red shoes) - 3/3 x 2 x 2

3/12=1/4 for bead bracelet

Red shoes - 1/3

Total probability for P(Bead bracelet and red shoes=1/4 x 1/3=1/12

P(Elastic bracelet and black shoes)

2/3 x 2 x 2

2/12=1/6 for elastic bracelet

Black shoes - 3/5

Total probablility for P(Elastic bracelet and black shoes)=1/6 x 3/5=3/10=1/10

Hope I helped!

5 0

3 years ago