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Final answer:

To find the dimensions of the picture, set up an equation using the length and width. Solve the equation to find the dimensions: length = 32 inches, width = 15 inches.

Explanation:

To find the dimensions of the picture, let's assume the length is x inches. According to the problem, the width is 17 inches shorter than the length, so the width would be (x - 17) inches. The total area of the picture is given as 480 square inches. We can set up an equation using the length and width: x(x - 17) = 480. Solve this quadratic equation to find the dimensions of the picture.

Expanding the equation, we get x^2 - 17x - 480 = 0. Factoring the quadratic equation, we find (x + 15)(x - 32) = 0. Therefore, the possible values for x are -15 and 32. Since the length cannot be negative, we discard -15. Thus, the length of the picture is 32 inches and the width is (32 - 17) = 15 inches.

Final answer:

The dimensions of the picture are 32 inches in length and 15 inches in width.

Explanation:

To find the dimensions of the picture, we need to establish equations based on the information provided. Let the length of the picture be L inches and the width be W inches.

Given that:

Plug the width expression from step 1 into the area equation from step 2 to form a quadratic equation:

L(L - 17) = 480

Solving this quadratic equation will give us the value of L. Once we have L, we can use the first equation to get W. Let's solve the quadratic equation:

L² - 17L - 480 = 0

Factoring the quadratic, we find that (L - 32)(L + 15) = 0. So L can be either 32 or -15. Since a length cannot be negative, L must be 32 inches, and W, therefore, is 32 - 17 = 15 inches.

The dimensions of the picture are therefore 32 inches by 15 inches.

Answer:

Equation is [tex]-8x+4y-12=0[/tex]

Step-by-step explanation:

An equation is a mathematical statement that two things are equal .

A system of equations is a set of two or more equations consisting of same unknowns.

Two equations [tex]a_1x+b_1y+c_1=0\,,\,a_2x+b_2y+c_2=0[/tex] have unique solution if [tex]\frac{a_1}{a_2}\neq \frac{b_1}{b_2}[/tex]

infinite solution if [tex]\frac{a_1}{a_2}= \frac{b_1}{b_2}=\frac{c_1}{c_2}[/tex]

no solution if [tex]\frac{a_1}{a_2}= \frac{b_1}{b_2}\neq \frac{c_1}{c_2}[/tex]

Here, given: [tex]-4x+2y=6[/tex]

We can write this equation as [tex]-4x+2y-6=0[/tex]

Take another equation as [tex]-8x+4y-12=0[/tex]

Here, [tex]a_1=-4\,,\,a_2=-8\,,\,b_1=2\,,\,b_2=4\,,\,c_1=-6\,,\,c_2=-12[/tex]

[tex]\frac{a_1}{a_2}=\frac{-4}{-8}=\frac{1}{2}\\\frac{b_1}{b_2}=\frac{2}{4}=\frac{1}{2}\\\frac{c_1}{c_2}=\frac{-6}{-12}=\frac{1}{2}[/tex]

such that [tex]\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}[/tex]

The current population of rabbits in the forest can be found by substituting t = 0 into the given formula P = 12,000 + 1000(t+1).

To find the number of rabbits currently in the forest, we can substitute the value of t = 0 into the given formula. The formula P = 12,000 + 1000(t+1) represents the population of rabbits in the forest at any given year, where t is the number of years from the present. Since we want to find the current population, we substitute t = 0:

P = 12,000 + 1000(0+1)

P = 12,000 + 1000(1)

P = 12,000 + 1000

P = 13,000

Therefore, there are currently 13,000 rabbits in the forest.

Answer:

There are so many little things that go into organic farming and each of them adds additional costs.

Some of these additional costs are listed in the article located here.

Use the data table at the end of the document to answer the questions.

Which cost category in the table shows an example of direct variation?  

Weed-Hand

The total cost for office expenses to run the organic strawberry farm varies directly with the number of months it is used. Which equation represents this scenario?

y = 63x

What would be the total cost for office expenses for 24 months?

$1,512.00

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80 pounds of fertilizer will be needed to cover 120 square feet of lawn.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

The relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

To write a proportion.

50/75 = x/120.

Then Cross multiply to get;

6,000/75x.

Now, divide each side by 75 to isolate the “x”.

x = 80.

Therefore 80 pounds of fertilizer will cover 120 square feet of lawn.

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We have that the concentration of the nitrate ions  is mathematically given as

M= 0.250M

Question Parameters:

Generally the equation for the Reaction   is mathematically given as

Mg(NO₃)₂ --> Mg²⁺ + 2NO₃⁻

Each mole of the Mg(NO₃)₂ will dissociate into 2 moles of NO₃⁻.

the concentration of the nitrate ions is    

M = (0.125M)(2)

M= 0.250M

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After traveling a distance of 2π/3 on the unit circle in the counter-clockwise direction from (1,0), the ant stops at the coordinates (-1/2, √3/2), which are found using the unit circle's parametric equations.

The question asks about finding the coordinates of a point on the unit circle after traveling a certain distance in the counter-clockwise direction. The distance given is 2π/3, which, when considering the circumference of the unit circle (2π), represents an arc that is a third of the full circle. The unit circle is centered at the origin (0,0) and has a radius of 1, with points on it defined by cos(θ) and sin(θ) for any angle θ measured in radians from the positive x-axis.

Traveling 2π/3 radians from (1,0) in the counter-clockwise direction lands us at an angle of 2π/3 radians from the positive x-axis. The coordinates of the point on the unit circle at this angle can be found using the circle's parametric equations x = cos(θ) and y = sin(θ). Substituting θ with 2π/3, we obtain:

Therefore, the coordinates of the point where the ant stops are (-1/2, √3/2).

Like terms are terms with the same variables raised to the same powers, allowing them to be combined by addition or subtraction.

Terms that have the same variables raised to the same powers are called like terms.

In mathematics, particularly algebra, like terms are terms that contain the same variables to the same power, which allows them to be combined through addition or subtraction. An example of like terms would be 3x2y and 7x2y. Both contain the variable x raised to the second power, and y raised to the first power, thus they can be combined to form 10x2y.

Terms that have the same variables raised to the same powers are called "like" terms. In algebraic expressions, like terms are essential for simplifying and combining expressions efficiently. The concept is rooted in the understanding that terms with identical variable factors, such as x^2 or y, can be grouped together. This simplification aids in solving equations, factoring, and performing various algebraic operations.

Identifying and combining like terms streamline mathematical processes, providing a clearer and more concise representation of expressions. Ultimately, the recognition of like terms is a fundamental aspect of algebraic manipulation, contributing to the coherence and simplicity of mathematical expressions.

Final answer:

Terms that have identical variables raised to the same powers are called like terms. They can be added or subtracted as they represent the same quantities of variables to the same exponent powers.

Explanation:

Terms that have the same variables raised to the same powers are called like terms.

Like terms can be combined (added or subtracted) because they represent the same types of quantities raised to the same power. For example, in the expression 3x2 + 5x2, both terms are like terms because they both involve the variable x raised to the second power, which means that these two terms can be added together to get 8x2.

Raising a number to an exponent, such as squaring (raising to the second power) or cubing (raising to the third power), is a way to express repeated multiplication of the number by itself. For instance, the expression 52 = 5 x 5 = 25 shows that 5 is being squared, and the result is 25.

57 / 3 =19

18 + 19 +20 = 57

 largest number is 20

Answer:

20

Step-by-step explanation:

he  has 4 pennies

2 nickels=10 cents

3 dime=30 cents

             20 cents

40 +4 pennies =44 cents

Answer:

Keith has 4 pennies, 3 dimes and 2 nickels.

Step-by-step explanation:

Let the pennies be = p

Let the nickels be = n

Let the dimes be = d

Total coins Keith has = 9

So, 1st equation form :

[tex]p+n+d=9[/tex]     ....(1)

Keith has one more dime than nickels, means [tex]d=n+1[/tex]   .....(2)

Total value of these coins is $0.44

So, third equation forms:

[tex]0.01p+0.05n+0.10d=0.44[/tex]   .....(3)

Substituting (2) in (1)

[tex]p+n+n+1=9[/tex]

[tex]p+2n=8[/tex]     .....(4)

Similarly substitution (2) in (3)

[tex]0.01p+0.05n+0.10(n+1)=0.44[/tex]

[tex]0.01p+0.05n+0.10n+0.10=0.44[/tex]

[tex]0.01p+0.15n=0.34[/tex]    ....(5)

Now multiplying (4) by 0.01 and subtracting by (5)

[tex]0.01p+0.02n=0.08[/tex] subtracting this from (5) we get

[tex]0.13n=0.26[/tex]

So, we get n = 2

[tex]p+2n=8[/tex]

[tex]p+2(2)=8[/tex]

[tex]p+4=8[/tex]

p = 4

[tex]p+n+d=9[/tex]

[tex]4+2+d=9[/tex]

[tex]6+d=9[/tex]

d = 3

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Therefore, Keith has 4 pennies, 3 dimes and 2 nickels.

Answer:

A. yes...it is a function because there is no repeating x values.

B. (5,2)(6,5)...slope = (5 - 2) / (6 - 5) = 3

y = mx + b

slope(m) = 3

(5,2)...x = 5 and y = 2

now we sub

2 = 3(5) + b

2 = 15 + b

2 - 15 = b

-13 = b

so the equation is : y = 3x - 13

when x = 7

y = 3(7) - 13....y = 21 - 13....y = 8

f(x) = 2x + 13...when x = 7

f(7) = 2(7) + 13...f(7) = 14 + 13.....f(7) = 27

the function has a greater value

C. f(x) = 2x + 13....when f(x) = 75

75 = 2x + 13

75 - 13 = 2x

62 = 2x

62/2 = x

31 = x

Step-by-step explanation:

To make 6 batches of muffins, George needs 3 cups of milk, which is found by multiplying the ratio of 1/2 cup of milk per batch by 6.

The question is asking to determine the quantity of an ingredient needed for a recipe, specifically milk for pancakes, according to a given ratio. The question is based on a proportion where 1/2 cup of milk is needed for one batch of pancakes. To find out how many cups of milk are needed for 6 batches of muffins (assuming muffins and pancakes require the same amount of milk which appears to be a minor typo in the scenario), we multiply the 1/2 cup required for one batch by 6.

Thus, the calculation is: 1/2 cup of milk × 6 batches = 3 cups of milk.

To find out how many cups of milk George needs to make 6 batches of muffins, we first need to determine the number of cups needed for one batch. Let's assume that each batch requires 3 cups of milk. Therefore, to make 6 batches, George would need 3 cups x 6 batches = 18 cups of milk

To find the volume of a cube with an edge length that is increasing with time, express the edge length as a function of time and substitute this into the volume formula. In the given problem, the edge length is increasing 2 feet per minute, thus the volume V of the cube at any given time can be expressed as V = (4 + 2t)³.

The volume of a cube is calculated by raising the edge length to the power of three (V = e³). For this problem, we know that the edge e is increasing at a rate of 2 feet per minute. Therefore, we can express e as a function of time t, where e = 4 + 2t. Substituting this back into our volume equation, we receive the volume V of the cube as a function of time t: V = (4 + 2t)³.

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