Find the volume of the triangular prism.

A. 120 yd3
B. 130 yd3
C. 240 yd3
D. 260 yd3

To find the dimensions of the rectangular corral split into two pens that produce the greatest enclosed area, use the perimeter and area formulas for a rectangle.

To find the dimensions of the rectangular corral split into two pens that produce the greatest enclosed area, we can use the perimeter formula for a rectangle. Let L and W represent the length and width of each pen. The total fencing length will be 2L + 3W, and since it is given as 630 feet, we have 2L + 3W = 630. We can rearrange this equation to solve for one variable in terms of the other, substitute it into the area formula for a rectangle, and then find the maximum value using the first derivative test.

Using the equation 2L + 3W = 630, we can solve for L in terms of W: L = (630 - 3W) / 2. Substituting this into the area formula A = LW, we get A = (630 - 3W)W / 2. Taking the first derivative of A with respect to W, we have dA/dW = 630/2 - 3W/2. Setting this equal to zero, we find W = 105. Substituting this back into the equation 2L + 3W = 630, we can solve for L: 2L + 3(105) = 630, and simplifying gives L = 210/2 = 105.

Therefore, the dimensions of each pen that produce the greatest enclosed area are 105 feet by 105 feet.

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Final answer:

Two functions are inverses if, when composed, they return the original input value (identity function). For example, an exponential function and its inverse, the natural log, undo each other. This can be checked with the equations by rearranging and setting equal to isolate constants or applying the functions one after the other.

Explanation:

How to Determine if Two Functions Are Inverses

To determine if two functions, F and G, are inverses of each other, you can compose the functions and see if the result is the identity function. The composition of two inverse functions will output the original input value, meaning that if you take a value x, apply function F to get F(x), and then apply function G to this result, i.e., G(F(x)), you should get x back if F and G are true inverses. The same must hold true in the reverse order, where G is applied first, followed by F: F(G(x)) = x.

To illustrate this concept, let's consider the exponential function and its inverse function, the natural log.Exponential functions are undone by their inverses, natural logs, according to the relation[tex]eln(x) = ln(ex)[/tex]to see this is with the equations involving a constant A, for instance, these can be rearranged to isolate ln A and set them equal to obtain an identity.

Another familiar context is the square and square root functions. To "undo" a square, you take the square root and vice versa. This "undoing" operation is a key characteristic of inverse functions.

Remember that not all functions have inverses, only those that are one-to-one, meaning for each x there is a unique y and vice versa, have inverses that can be defined over their entire domain.

You would need  1/12 cups of water in order to make only 1/6 of the recipe

A fraction is defined as a numerical representation of a part of a whole that represents a rational number.

Arithmetic operations can also be specified by addition, subtraction, dividing, and multiplying built-in functions.

When making bread from scratch, the recipe calls for 2/4 cup of water. If you need to make a smaller portion of the recipe,

If a recipe requires 2/4 cup of water, then a fraction of the recipe requires 1/6 cup of water.

So the fraction of the recipe requires 1/6 cup of water as

⇒ 1/6×2/4

⇒  2/24

⇒  1/12

Hence, you would need  1/12 cups of water in order to make only 1/6 of the recipe.

Learn more about the fraction here:

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Answer:

3*4*7

Step-by-step explanation:

There is total of 84 combinations

To find the square root of an improper fraction, simplify the fraction if possible, convert it into perfect squares, or use the properties of exponents. If it is already in simple form like 49/100, directly find the square roots of the numerator and denominator separately. Otherwise, use multiplication to manipulate the fraction into a more manageable form for taking square roots.

To find the square root of an improper fraction, the process involves several steps, similar to finding the square root of any number. First, simplify the fraction if possible. In some cases, this may involve multiplying the numerator and denominator by the same factor to make the radical easier to simplify. For instance, if you have an improper fraction like 49/100, you can easily identify that both the numerator and the denominator are square numbers. So, the square root of 49/100 equals the square root of 49 divided by the square root of 100, which simplifies to 7/10.

When the fraction cannot be simplified easily, a different approach is by converting the numerator and denominator into perfect squares, simplifying the expression within the radical, or using properties of exponents to re-express the square root. For example, if you have a fraction like 1/2, you can multiply both the numerator and denominator by 2 to get 2/4. Then you would rewrite the square root as the square root of 2 over the square root of 4, which simplifies to the square root of 2 over 2.

3 inside angles need to = 180, since it is a right triangle, the bottom left angle = 80 so the other 2 angles need to equal 90 degrees

 so you have:

(8x+2) + (9x+3) = 90

 combine like terms to get:

17x+5 = 90

subtract 5 from both sides:

17x = 85

 divide both sides by 17

x = 85/17 = 5

x = 5

The correct answer is 24.

Answer:

(C) [tex]{\angle}MOP=61^{\circ}[/tex]

Step-by-step explanation:

Given: It is given that angle MON is a straight angle and OP is bisects angle MOQ.

To find: The measure of the angle MOP.

Solution: It is given that angle MON is a straight angle and OP is bisects angle MOQ.

Thus, using the straight line property, we get

[tex]{\angle}MOP+{\angle}POQ+{\angle}QON=180^{\circ}[/tex]

[tex]2{\angle}MOP+{\angle}QON=180^{\circ}[/tex]

Substituting the given values, we get

[tex]2{\angle}MOP+58^{\circ}=180^{\circ}[/tex]

[tex]2{\angle}MOP=122^{\circ}[/tex]

[tex]{\angle}MOP=61^{\circ}[/tex]

Hence, option (C) is correct.

Answer: C - 61°

Step-by-step explanation: edge

[tex]\large\text{Hey there!}[/tex]

[tex]\large\text{Step 1: You have to \underline{\bf{CROSS MULTIPLY}}}[/tex]

[tex]\mathsf{\dfrac{14}{x}=\dfrac{56}{16}}\\\\\mathsf{14\times16=56\times x}\\\\\mathsf{14(16)= \ \bf{\underline{224}}}\\\\\mathsf{56(x)=\underline{\bf{56x}}}[/tex]

[tex]\huge\text{We get: \bf{\underline{224 = 56x}}}[/tex]

[tex]\large\text{Step 2: }\large\text{\bf{\underline{FLIP}}}\large\text{ the new equation around}[/tex]

[tex]\huge\text{\bf{\underline{56x = 224}}}[/tex]

[tex]\large\text{Step 3: \bf{\underline{DIVIDE}}}\large\text{ both of your sides by 56}[/tex]

[tex]\mathsf{\dfrac{56x}{56}=\dfrac{224}{56}}\\\\\mathsf{Cancel\ out: \dfrac{56x}{56}\ because\ that\ gives\ us\ 1}\\\\\mathsf{Keep: \dfrac{224}{56}\ because\ it\ gives you\ the\ value\ of\ x }\\\\\mathsf{\dfrac{224}{56}=4}\\\\\mathsf{x=4}[/tex]

[tex]\boxed{\large\text{So, the reason why x = 4 for this equation is because } \mathsf{\dfrac{14}{4}}\ \& \ \dfrac{56}{16}\ \large\text{ are equal to each}}\\\boxed{\large\text{other. }}[/tex]

[tex]\mathsf{\dfrac{14}{4} = 3.5}[/tex]

[tex]\mathsf{\dfrac{56}{16} = 3.5}[/tex]

[tex]\huge\text{\bf{3.5 = 3.5}} \ \checkmark[/tex]

[tex]\large\text{Good luck on your assignment and enjoy your day!}[/tex]

~[tex]\dag\dfrac{\frak{LoveYourselfFirst}}{:)}[/tex]

Final answer:

Tom would need to rent the basketball court 15 times for the gym membership to be more cost-effective than paying the non-member rental fee.

Explanation:

To determine how many times Tom would need to rent the basketball court to make the membership cheaper than being a non-member, we can set up a cost equation and find the breakeven point. Let's assume that Tom rents the court x times. For a non-member, the cost is $11 each time, which results in 11x dollars. For a member, the cost includes the one-time membership fee of $135 plus $2 each time the court is rented, summing to 135 + 2x dollars.

To find the breakeven point, we equate the two costs and solve for x:

11x = 135 + 2x

9x = 135

x = 135 / 9

x = 15

Thus, Tom would need to rent the court 15 times for the membership cost to be cheaper than paying as a non-member.

Final answer:

The second and third terms of the arithmetic sequence are -12 and -17, found by adding the common difference of 5 to each succeeding term in the sequence.

Explanation:

The question is asking to find the second and third terms of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant. This constant difference is known as the common difference. In the given arithmetic sequence (-7, _, _, -22, -27), we can find the common difference by subtracting two successive terms. For this sequence, the difference between -27 and -22 is -27 - (-22) = -27 + 22 = -5. This implies that to find the previous terms, we should add 5 to each term going backwards.

Starting with -22 and adding 5, we get -17 for the third term. Similarly, adding 5 again for the second term, we get -12. Therefore, the second and third terms of the arithmetic sequence are -12 and -17, respectively.

Answer:  The correct option is (B) 3.

Step-by-step explanation:  We are given to find the length of the leg s in the triangle shown in the figure.

From the figure, we note that

the given triangle is a right-angled one, where

length of the hypotenuse, h = √18 units,

length of one leg, p = 3 units

and

length of the other leg, s = ?.

From Pythagoras theorem, we have

[tex]h^2=p^2+s^2\\\\\Rightarrow (\sqrt{18})^2=3^2+s^2\\\\\Rightarrow 18=9+s^2\\\\\Rightarrow s^2=18-9\\\\\Rightarrow s^2=9\\\\\Rightarrow s=3.[/tex]

Thus, the length of the leg s is 3 units.

Option (B) is CORRECT.

Answer: [tex]3x+4y=16[/tex]

Step-by-step explanation:

Given equation of a line = [tex]3x + 4y = 15[/tex]

i.e. [tex]4y = 15-3x[/tex]

i.e. [tex]y=\dfrac{15}{4}-\dfrac{3}{4}x[/tex]

i.e. [tex]y=-\dfrac{3}{4}x+\dfrac{15}{4}[/tex]  (1)

It is in intercept form [tex]y=mx+c[/tex]   (2) , where m is slope.

By comparing (1) and (2) the slope of line : [tex]m=-\dfrac{3}{4}[/tex]

Also the parallel lines have same slope .

The equation of a line passing through a point (a,b) and having a slope of 'm' is given by :-

[tex](y-b)= m(x-a)[/tex]

Then the equation of the line that passes through (8,-2) and having slope of [tex]m=-\dfrac{3}{4}[/tex] is given by :-

[tex](y-(-2))=- \dfrac{3}{4}(x-8)[/tex]

[tex]4(y+2)=-3(x-8)[/tex]

[tex]4y+8=-3x+24[/tex]

[tex]3x+4y=24-8[/tex]

[tex]3x+4y=16[/tex]

Hence , the  equation of a line that passes through the point (8, −2) and is parallel to the line whose equation is [tex]3x + 4y = 15[/tex] is [tex]3x+4y=16[/tex]

.

Final answer:

The question involves using the Pythagorean theorem to find the missing side 'X' in a right triangle with sides 'X', 4, and 7. 'X' is calculated by rearranging the theorem to X² = c² - b² and plugging in the known values. The equation simplifies to X = √33.

Explanation:

The student is asking for an equation to solve for the missing side of a right triangle, where the sides are given as 'X', 4, and 7. The Pythagorean theorem, which states that a² + b² = c², is the appropriate tool for solving this problem. Since the triangle is right-angled, we assume 'X' is one leg, 4 is the other leg, and 7 is the hypotenuse as it is the longest side.

To find 'X', we rearrange the theorem to solve for the unknown leg (X) as follows: X² = c² - b². Plugging the values into the equation gives us X² = 7² - 4² which simplifies to X² = 49 - 16, hence X² = 33. Consequently, X can be found by taking the square root of 33, which we rewrite as X = √33.

To find the value of 'x' that makes the function r(x) = 19, we need to solve the equation -5x - 1 = 19 for 'x'. After doing so, we obtain x = -4.

This question involves finding the value of 'x' in a mathematical function provided, in order to get a specific output. The function provided is r(x) = -5x - 1 and the question states that the output r(x) should be 19. We can setup the equation -5x - 1 = 19 and solve for 'x'.

Adding 1 to both sides of the equation, we obtain -5x = 20. Dividing both sides by -5 gives x = -4. So, the value of 'x' that makes the function r(x) = 19 is x = -4.

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