the right triangular prism and the square rectangular prism both haven't had a 10 in in the same volume. A plane slices both solid parallel to their bases. The base of the triangular prism is a right triangle with legs 8 and 9 in Long. Find the length of the side of the square prism's cross-section in inches.​

Integration

Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Double Integrals

Polar Coordinates Conversions:

Integral Conversion [Polar Coordinates]:
[tex]\displaystyle \iint_T {f(x, y)} \, dA = \iint_R {f(r, \theta)r} \, dr \, d\theta[/tex]

The formatting of the question was thrown off, so I have defined it down below.

We are given an integral and asked to convert to polar coordinates as well as evaluate it:
[tex]\displaystyle \int \limits^{8}_{-8} \int \limits^{\sqrt{64 - x^2}}_{0} \sin(x^2 + y^2) \, dy \, dx[/tex]

It would be quite difficult to evaluate the given integral using conventional methods, so we apply polar conversion to evaluate the integral. Let's start out by converting the function and the bounds.

[Bounds] Cartesian to Polar:

[tex]\displaystyle \left \{ {{-8 \leq x \leq 8} \atop {0 \leq y \leq \sqrt{64 - x^2}}} \right \longrightarrow \left \{ {{0 \leq r \leq 8} \atop {0 \leq \theta \leq \pi}} \right[/tex]

[Function] Cartesian to Polar:

[tex]\displaystyle f(x ,\ y) = \sin(x^2 + y^2) \longrightarrow f(r ,\ \theta) = \sin r^2[/tex]

Now that we've converted to polar coordinates, we can convert the integral using our integral conversion listed under "Multivariable Calculus":

[tex]\displaystyle \int \limits^{8}_{-8} \int \limits^{\sqrt{64 - x^2}}_{0} \sin(x^2 + y^2) \, dy \, dx \longrightarrow \int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta[/tex]

We can now evaluate the polar integral using basic integration techniques listed under "Calculus":

[tex]\displaystyle \begin{aligned}\int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta & = \int \limits^{\pi}_{0} \underbrace{\int \limits^{8}_{0} r \sin r^2 \, dr \, }_{u = r^2 ,\ du = 2r \, dr} d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \int \limits^{8}_{0} 2r \sin r^2 \, dr \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \int \limits^{r = 8}_{r = 0} \sin u \, du \, d\theta \\\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta & = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( - \cos u \bigg) \bigg| \limits^{r = 8}_{r = 0} \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( - \cos r^2 \bigg) \bigg| \limits^{r = 8}_{r = 0} \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( - \cos 64 + 1 \bigg) \, d\theta \\& = \frac{1}{2} \int \limits^{\pi}_{0} \bigg( 1 - \cos 64 \bigg) \, d\theta \\\end{aligned}[/tex]

[tex]\displaystyle \begin{aligned}\int \limits^{\pi}_{0} \int \limits^{8}_{0} r \sin r^2 \, dr \, d\theta & = \frac{1}{2} \bigg(1 - \cos 64 \bigg) \bigg( \theta \bigg) \bigg| \limits^{\pi}_{0} \\& = \boxed{ \frac{\pi}{2} \bigg( 1 - \cos 64 \bigg) }\end{aligned}[/tex]

∴ the integral equals:
[tex]\displaystyle \boxed{ \frac{\pi}{2} \bigg( 1 - \cos 64 \bigg) }[/tex]

[tex]\displaystyle \int \limits^{8}_{-8} \int \limits^{\sqrt{64 - x^2}}_{0} \sin(x^2 + y^2) \, dy \, dx = \boxed{ \frac{\pi}{2} \bigg( 1 - \cos 64 \bigg) }[/tex]

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Topic: Multivariable Calculus

Unit: Double Integrals

Step-by-step explanation:

(i) Sum the forces at point M in the x direction.

∑F = ma

-T sin 30° − T sin 30° + 5 N = 0

2T sin 30° = 5 N

T = 5 N

(ii) Sum the forces on the ring in the x direction.

∑F = ma

T sin 30° − N = 0

N = 2.5 N

Sum the forces on the ring in the y direction.

∑F = ma

T cos 30° − mg − Nμ = 0

Nμ = T cos 30° − mg

2.5 μ = 5 cos 30° − 2

μ = 0.932

(iii) Sum the forces on the ring in the y direction.

∑F = ma

T cos 30° − m₁g − m₂g + Nμ = 0

m₂g = T cos 30° − m₁g + Nμ

m (10) = 5 cos 30° − 2 + (2.5)(0.932)

m = 0.466 kg

We have,

Now,

∠AOB + ∠BOC = ∠A0C

⇒ x + 128° = 180°

⇒ x = 180° - 128°

⇒ x = 52°

Answer:

Irrational, non-terminating

Step-by-step explanation:

This number is irrational because it is a non-terminating decimal. Notice how it continues with the '...'. <3

Answer:

-1.5

Step-by-step explanation:

Answer:

see below

Step-by-step explanation:

-1.5

-1 1/2

-6/4

Answer:

11

Step-by-step explanation:

Just multiply 4 times 2 3/4 and youll get 11

If one cup of soy milk contains 4g total fat then [tex]2\frac{3}{4}[/tex] cups of soy milk contains 11g of  total fat.

             [tex]2\frac{3}{4}[/tex] cups contains x grams of total fat.

         ⇒ x  =  4([tex]\frac{11}{4}[/tex])

         ⇒ x  = 11 grams

     Hence 11 grams of total fat is there in [tex]2\frac{3}{4}[/tex] cups of soy milk.

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Answer:

$3,636.70

Step-by-step explanation:

Given the following:

Initial price of Gold (A) = $2500

Rate of appreciation (r) = 5.5% = 0.055

Worth of gold in 7 years will be?

Period (p) = 7

Using the compound interest formula :

Let F = final amount

F = A( 1 + r/n)^nt

n = number of times Appreciation occurs per period.

Since rate compounds yearly, then, n = 1

F = $2500( 1 + 0.055/1)^(7*1)

F = $2500(1 + 0.055)^7

F = $2500(1.055)^7

F = $2500(1.454679161133794609375)

F = $3636.6979

Amount in 7 years = $3,636.70

Answer:

What lined graph?

Step-by-step explanation:

Answer:

Please give a lined graph. However, if your question is like mine, I may be able to help.

Here, the answer is D. For the slope, remember: Rise over Run. The slope will be a fraction, so let the amount of points between points on the line (If that makes sense) be that fraction. The amount between points vertically is the Numerator (top) of your fraction, and the amount horizontally is the denominator (Bottom). If the Denominator is 1, your fraction is a whole number.

Answer:

84

Step-by-step explanation:

The least common denominator of the 7, 6, 12, which are the denominators of the fractions in the given linear equation, is the least expression or number that is divisible by 7, 6, 12.

To find the LCD, express each number as factors of itself as follows:

[tex] 7 = 1*7 [/tex]

[tex] 6 = 2*3 [/tex]

[tex] 12 = 2^2*3 [/tex]

Find the product of the highest terms

The product = LCD = [tex] 1*2^2*3*7 = 84 [/tex]

LCD of 7, 6, 12 = 84

Answer:

84

Step-by-step explanation:

Answer:

Translation

Step-by-step explanation:

The object is only moved, which is what happens when a translation occurs

Answer:

The two iterations of f(x) = 1.5598

Step-by-step explanation:

If we apply  Newton's iterations method, we get a new guess of a zero of a function, f(x), xₙ₊₁, using a previous guess of, xₙ.

xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ)

Given;

f(xₙ) = cos x, then  f'(xₙ) = - sin x

cos x / - sin x = -cot x

substitute in "-cot x" into the equation

xₙ₊₁ = xₙ - (- cot x)

xₙ₊₁ = xₙ + cot x

x₁ = 0.7

first iteration

x₂ = 0.7 + cot (0.7)

x₂ =  0.7 + 1.18724

x₂ = 1.88724

second iteration

x₃ = 1.88724 + cot (1.88724)

x₃ = 1.88724 - 0.32744

x₃ = 1.5598

To four decimal places = 1.5598

Answer:

x = 5

Step-by-step explanation:

0.5x + 3.5 = 6

0.5x = 6 - 3.5

0.5x = 2.5

x = 2.5 / 0.5

x = 5

Answer:

Step-by-step explanation:

f(x) = x² - 2x - 24

g(x) = x + 4

To find (f-g)(x) subtract g(x) from f(x)

That's

(f-g)(x) = x² - 2x - 24 - ( x + 4)

(f-g)(x) = x² - 2x - 24 - x - 4

Group like terms

We have

(f-g)(x) = x² - 2x - x - 24 - 4

Simplify

We have the final answer as

Hope this helps you

Answer:

3L

Step-by-step explanation:

Perimeter is the sum of side lengths

Since the triangle is equilateral, it has all sides equal and each has length of L

So perimeter is:

Answer:

4 OR 4.5 SECONDS

Step-by-step explanation:

Answer: 3000 in^2

Explanation:

Answer:

m<3 + m<4:= 180°

m<2 + m<4 + m<6 = 180°

m<2 + m<4 = m<5

Step-by-step explanation:

<3 and <4 are linear pairs. They are angles on a straight line. Angles in a straight line sum up to give 180°. Therefore, the statement "m<3 + m<4:= 180°" is TRUE.

<2, <4, <6 are interior angles of a triangle. The sum of the angles in a ∆ = 180°. Therefore, the statement, "m<2 + m<4 + m<6 = 180°" is TRUE.

<2 and <4 are opposite interior angles of the ∆, while <5 is an exterior angle to the ∆. Based on the external angle theorem of a ∆, the statement, "m<2 + m<4 = m<5" is TRUE.

<1 and <2 are a linear pair, and are angles on a straight line. Their sum cannot give us 90°.

m<4 + m<6 ≠ m<2. Rather, 180 - (m<4 + m<6) = m<2 (sum of angles in a ∆)

m<2 + m<6 ≠ m<5. Rather, m<2 + m<4 = m<5

The correct equations are:

m∠5 + m∠6 = 180°

m∠2 + m∠3 =  m∠6

m∠2 + m∠3 + m∠5 = 180°

m∠2 + m∠5 =  m∠4

Triangle is a polygon with three angles and three sides. The sum of angles in a triangle is 180 degree.

From the diagram:

m∠5 + m∠6 = 180° (angle in a straight line)

But:

m∠2 + m∠3 + m∠5 = 180

m∠2 + m∠3 + m∠5 = m∠5 + m∠6

m∠2 + m∠3 =  m∠6

Also:

m∠2 + m∠3 + m∠5 = 180° (sum of angles in a triangle)

But:

m∠3 + m∠4 = 180° (angle in a straight line)  

m∠2 + m∠3 + m∠5 = m∠3 + m∠4

m∠2 + m∠5 =  m∠4

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Answer:

6

Step-by-step explanation:

A) f(x+h)=6(x+h)-6

         =6x+6h-6

B)(6x+6h-6)-(6x-6)

   6x+6h-6-6x+6

    =>6h

C=6h/h

C=6

★ Solution :

[tex]:\implies\sf 3(4x - 7) = 27\:\:\:\:\Bigg\lgroup \bf{Given\: Equation}\Bigg\rgroup \\\\\\:\implies\sf 12x - 21 = 27\\\\\\:\implies\sf 12x = 27 + 21\\\\\\:\implies\sf 12x = 48\\\\\\:\implies\sf x = \dfrac{48}{12}\\\\\\:\implies\underline{\boxed{\sf x = 4}}[/tex]

Answer:

[tex]3(x - 7) = 27[/tex]

[tex]⟹3x - 21 = 27[/tex]

[tex]⟹3x = 27 + 21[/tex]

[tex]⟹3x = 48[/tex]

[tex]⟹x = \frac{48}{3} [/tex]

[tex]⟹x = 16[/tex]

Answer:

5

Step-by-step explanation:

Given that each zodiac sign occupies 1/12 of a year.

Then the minimum number of persons for Y[all different signs] < 0.5,

The probability of at least two having the same sign is 1 minus the probability of all having different signs.

This can be represented as A [at least 2 person share the same sign] = 1 - Y[all different signs] must be > 0.5

Therefore we have 1 - 12/12 *11/12 * 10/12 *9/12 *8/12 = 0.38

This implies that the lowest number will be found to be 5

Hence, the correct answer is 5.

Answer:

(4,7)

Step-by-step explanation:

f(x)=5x-13

Let this equal to 7

7=5x-13  

Add 13 to each side

7+13 = 5x-13+13

20 = 5x

Divide by 5

20/5 = 5x/5

4 =x

The ordered pair is

(4,7)

Answer:

10 miles

Step-by-step explanation:

3a/5 + 4 = a

a = race long

4 = a - 3a/5

4 = 5a/5 - 3a/5

4 = 2a/5

4*5/2 = a

a = 10

The length of the entire race is 10 miles.

Let length of race = r

Miles left after running 3/5 of race :

We can solve the expression thus :

r - 3r/5 = 4

r - 3r = 4 * -5

r - 3r = -20

-2r = -20

divide both sides by -2 to isolate r

r = 10

Hence, the length of the entire race is 10 miles.

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Answer:

the vetrex

Step-by-step explanation:

Answer: y= -2.5

Step-by-step explanation:

Distribute the -2 and 3 into the parentheses. -6 + 6 cancels out so you are left with 3y on the right side. Add 2y to the right side leaving you with -2=5y. Divide both sides by -2

Answer:

Step-by-step explanation:

[tex] \sf{ - 2(y + 1) = 3(y - 2) + 6}[/tex]

Distribute -2 through the parentheses

Similarly, Distribute 3 through the parentheses

⇒[tex] \sf{ - 2y - 2 = 3y - 6 + 6}[/tex]

Since two opposites add up to zero , remove them

⇒[tex] \sf{ - 2y - 2 = 3y}[/tex]

Move 3y to Left hand side and change it's sign

Similarly, move -1 to right hand side and change it's sign

⇒[tex] \sf{ - 2y - 3y = 2}[/tex]

Collect like term

⇒[tex] \sf{ - 5y = 2}[/tex]

Divide both sides of the equation by -5

⇒[tex] \sf{ \frac{ - 5y}{ -5 } = \frac{2}{ - 5} }[/tex]

Calculate

⇒[tex] \sf{ - 0.4}[/tex]

Hope I helped!

Best regards!

Answer:

The true statements are;

(i) m - n = n - m

(ii) +m (-n) = m - n

(iii) 0 = m - n

Step-by-step explanation:

We are given that m and n are both integers and we have to select all the statements that are true if m and n are also equal to each other.

(i) The given situation is: m - n = n - m

LHS = m - n

       = m - m   {because m and n are equal}

       = 0

RHS = n - m

       = n - n   {because m and n are equal}

       = 0

Hence, the given statement is true because LHS = RHS = 0.

(ii) The given situation is: +m (- n) = m - n

LHS = +m (- n)

       = m - n

       = m - m  {because m and n are equal}

       = 0

RHS = m - n

       = n - n   {because m and n are equal}

       = 0

Hence, the given statement is true because LHS = RHS = 0.

(iii) The given situation is: 0 = m - n

LHS = 0

RHS = m - n

       = m - m   {because m and n are equal}

       = 0

Hence, the given statement is true because LHS = RHS = 0.

(iv) The given situation is: m + n = 0

RHS = 0

LHS = m + n

       = m + m   {because m and n are equal}

       = 2m

Hence, the given statement is not true because LHS [tex]\neq[/tex] RHS.

#1) m - n = n - m

#2) +m (-n) = m - n

#3) 0 = m - n

hope this HELPS!!!!!!!

Answer:

7/16, 7/32, 7/64

Step-by-step explanation:

7, 7 /2 , 7/ 4 , 7/ 8

We multiply by 1/2 each time

7/8 *1/2 = 7/16

7/16*1/2 = 7/32

7/32 *1/2 = 7/64

Answer:

The answer is

Step-by-step explanation:

Since the triangle is a right angled triangle we can use trigonometric ratios to find y

To find y we use cosine

cos∅ = adjacent / hypotenuse

From the question

y is the adjacent

The hypotenuse is 15

So we have

[tex] \cos(35) = \frac{y}{15} \\ y = 15 \cos( 35 ) \\ y = 12.28728[/tex]

We have the final answer as

Hope this helps you

Answer:

4/9

Step-by-step explanation:

0.4444... can be written as a geometric series with first term 0.4 and common ratio 0.1.  Each new digit is 0.1 times the previous digit.

                               0.4

Then 0.4444... = ------------ = 0.4/0.9 = 4/9

                             1 - 0.1

You may check this result by dividing 4 by 9 on a calculator.