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3 years ago

Rewrite 2 7/9 as a improper fraction

Mathematics

1 answer:

Alborosie3 years ago

6 0

The answer would be 25/9 as a improper fraction

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What is 4,000,275,000 in word form?

castortr0y [4]

Answer:

Four billion, two-hundred seventy five thousand.

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3 years ago

Read 2 more answers

0.5y-2 2/5=1/2y+2.4 Help :l

lesya [120]

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3 years ago

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

Ben, Cam, and Justin are lumberjacks. The number of trees they chop down is given by b + 2 c + 3 j b+2c+3jb, plus, 2, c, plus, 3

ladessa [460]

Consider the complete question is "Ben, Cam, and Justin are lumberjacks. The number of trees they chop down is given by b+2c+3j, where b is the number of hours Ben spends chopping, c is the number of hours Cam spends chopping, and j is the number of hours Justin spends chopping. How many trees do they chop down after Ben spends 8 hours chopping, Cam spends 3 hours chopping, and Justin spends 4 hours chopping?"

Given:

Total number of trees chopped by them = $b+2c+3j$

b = number of hours Ben spends chopping = 8

c = number of hours Cam spends chopping = 3

j = number of hours Justin spends chopping = 4

To find:

The numerical value of total number of trees chopped by them.

Solution:

Total number of trees chopped by them = $b+2c+3j$

Put $b=8, c=3$ and $j=4$ in the given expression.

$Total=8+2(3)+3(4)$

$Total=8+6+12$

$Total=26$

Therefore, total number of trees chopped by them is 26.

8 0

3 years ago

Brandon just bought a new coffee table. The length of the coffee table is 5 more than 3 times the width. If the perimeter of the

a_sh-v [17]

Answer:

The length is 23 inches and the width is 6 inches.

Step-by-step explanation:

The perimeter for a rectangular shape is represented as:

P = 2L + 2W, where L represents length and W represents width

We can represent the length as:

L = 3W + 5

Substituting this into the perimeter function, we get:

P = 2 (3W + 5) + 2W

Substituting 58 for P, we get:

58 = 2 (3W + 5) + 2W

58 = 6W + 10 + 2W

58 = 8W + 10

58 - 10 = 8W + 10 - 10

48 = 8W

48 / 8 = 8W / 8

6 = W

With 6 being the established value for the width, we can substitute this back into the equation for length:

L = 3W + 5

L = 3(6) + 5

L = 18 + 5

L = 23

To check our work, we can substitute both the width and length into the perimeter equation:

P = 2L + 2W

58 = 2(23) + 2(6)

58 = 46 + 12

58 = 58

Therefore, length is 23 inches and the width is 6 inches.

5 0

3 years ago