A recipe for homemade modeling clay includes 1/3 cup of salt for every cup of water. If there are 6 cups of salt, how many gallons of water are needed? Identify the constant of proportionality.

Given the expressions, f(x)=√x+5, g(x)=3x-7, and h(x)=5x, the answer to f(g(5)) is 8. This is because when g(5) is substituted into f(x) we get f(g(5))=f(3x-7)=√(3x-7)+5. When 5 is substituted for x, this simplifies to f(g(5))=√(-2)+5 which equals 8.

The answer to g(f(78)) is 60. This is because when 78 is substituted into f(x) we get f(78)=√78+5 which simplifies to f(78)=9. When 9 is substituted into g(x) we get g(f(78))=3x-7 which simplifies to g(f(78))=3(9)-7 which equals 60.

Finally, the answer to h(g(f(2))) is 83. This is because when 2 is substituted into f(x) we get f(2)=√2+5 which simplifies to f(2)=3. When 3 is substituted into g(x) we get g(f(2))=3x-7 which simplifies to g(f(2))=3(3)-7 which equals 8. When 8 is substituted into h(x) we get h(g(f(2)))=5x which simplifies to h(g(f(2)))=5(8) which equals 83.

In summary, the answers to f(g(5)), g(f(78)), and h(g(f(2))) are 8, 60, and 83 respectively.

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Answer: Please see explanation column for answer.

x= -36

Step-by-step explanation:

Step 1

from the attachment, Carlo solving is given as

1/3 (1/2x- 9) = 1/2 (x+18)

1/6x-3= 1/2x+9

1/4x-3=9

1/4x=12

x=48

Carlos error started when he subtracted wrongly 1/2 x from 1/6x which gave him 1/4 x instead of - 1/3 x.

Solving correctly, we have that

1/3(1/2x-9) = 1/2(x+18)

1/6x -3= 1/2x+ 9

taking like terms to like terms

1/6x-1/2x= 9+3

(1-3)/6x= 12

-2/6x= 12

-1/3x= 12

multiplying both sides by -3

-1/3x X -3 = 12 X -3

x= -36

Answer:

slope = 1/2

Step-by-step explanation:

The slope formula:

The slope of a line that passes through points [tex] (x_1, y_1) [/tex] and [tex] (x_2, y_2) [/tex] is

[tex] slope = m = \dfrac{y_2 - y_1}{x_2 - x_1} [/tex]

To find the slope, we find two easy-to-read points. Easy-to-read points are usually on grid line intersections.

Points (0, 0) and (2, 1) are on grid line intersections and are easy to read.

We will now input the information from our two points into the slope formula to calculate the slope.

[tex] slope = m = \dfrac{1 - 0}{2 - 0} = \dfrac{1}{2} [/tex]

The line a and line c are perpendicular to each other.

Lines are known if they are parallel or perpendicular by there slopes.

Therefore, parallel lines have the same slopes and perpendicular have slopes that are negative reciprocals of one another.

The slope of a line is the change in the dependent variable to the change in the independent variable.

Therefore, let's find the slopes of each line.

line a:

slope = -3 - 1 / 6- 0  = - 4 / 6 = - 2 / 3

line b:

slope = -2 - 0 / 3 + 3 = - 2 / 6 = - 1 / 3

line c:

slope = 7 + 2 / 2 + 4 = 9 / 6 = 3 / 2

Therefore, line a and c are perpendicular

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The answers to all parts are shown below.

To determine the quantity or percentage of something in terms of 100, use the percentage formula. Per cent simply means one in a hundred. Using the percentage formula, a number between 0 and 1 can be expressed. A number that is expressed as a fraction of 100 is what it is. It is mostly used to compare and determine ratios and is represented by the symbol %.

Given:

7. CP of sound system = 499

loss= 45%

So, the selling price = 499 - 45% of 499

                                  = R 274.45

8. original price is =1084

The new price = 1084 + 8% of 1084

                                  = R 1,170.72

9. Percentage decreased

= ( 747 - 599) / 747 x 100

= 19.81%

10. Mandisa left with

= 1000 - (228.95 + 573.38)

= R 197.67

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Answer:

2.50x + 1.25y ≤ 30

x + y ≥ 14

x ≤ 10 and y ≤ 24

Step-by-step explanation:

we can by elimination if we multiply the second inequality by -2.5, we get:

-2.50x - 2.5y ≥ -35

if we add the first inequality to the above we get:

   2.50x + 1.25y ≤ 30

+  -2.50x - 2.5y ≥ -35

-1.25y = -5

divide each side by -1.25 to get:

y = 4

therefore, x = 10

if you graph each system you must look for the area where the shading of each inequality overlaps

upon graphing, the solution is x ≥ 10 and y ≤ 24

The problem involving exponents (1/5 x 5)⁵ is calculated to get 1

Information made available in the question is an equation of the form

= (1/5 x 5)⁵

The math question is solved using the idea of PEMDAS which is an acronym for order of arrangement of mathematical operations as

the full meaning include

P - parenthesis

E  - exponents

M - multiplication

D - division

A - addition

S - subtraction

applying the sequence the problem is solved using parenthesis first

= (1/5 x 5)⁵

= (1)⁵

Then exponents

= (1)⁵

= 1 * 1 * 1 * 1 * 1

= 1

The solution is 1

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In this case, the marked price of 220 is the cost price and discount of 20% is announced on sales, so the selling price is 176.

The t-shirt costs $166.5 to purchase. To do this, multiply the listed price (220) by the discount rate (20%) stated as a decimal. This will give you the amount of the discount (0.20).

220*0.20 =44

The selling price is then determined by deducting the discount (44) from the marked price (220):

220-44 =176

Before applying any markups or reductions, the retailer paid the item's cost price, commonly referred to as the wholesale price. The price a consumer pays to purchase an item is known as the selling price. The selling price will be less than the advertised price if a discount is being offered, and the difference between the two represents the discount.

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The number √5 + √16 is an irrational number

From the question, we have the following parameters that can be used in our computation:

\sqrt{5}+\sqrt{16}

Express properly

So, we have the following representation

√5 + √16

Take the square root of 16

This gives

√5 + 4

Rewrite as

4 + √5

The above expression is an irrational number

This is so because of the radical symbol on 5

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An inequality representing c, the total number of 150-kilogram crates that can be loaded into the shipping container: c ≤ 98

We know that an inequality is a mathematical statement that represents the inequalities between two algebraic expressions with two variables .

It is an unequal relation between two algebraic expressions.

Here, the greatest weight that can be loaded into the container = 26,000 kilograms

the weight of each crate = 150 kilogram

and the weight of other shipment = 11300 kilograms

Let c be the total number of 150-kilogram crates that can be loaded.

So, we get an inequality:

150 c + 11300 ≤ 26,000

150 c ≤ 26,000 - 11300

150 c ≤ 14700

c ≤ 98

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The zeros of the function are -10 and +10

lesser x = -10

greater x = +10

The zeros of a quadratic equation are the points where the graph of the quadratic equation crosses the x-axis. . The zeros of quadratic equation will always have two values.

f(x) = 8x²- 800

f(x) = y

y = 8x²-800

when the graph crosses x axis the value of y = 0

therefore 0 = 8x²-800

divide both sides by 8

0 = x² - 00

x² = 100

x= √100

x= ±10

therefore lesser x = -10

greater 10 = +10

therefore the zeros of the quadratic equation f(x) = 8x²- 800 = +10 and -10

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Answer:

[tex] \sf \: x = 5[/tex]

Step-by-step explanation:

Now we have to,

→ Find the required value of x.

The equation is,

→ 15 - x = 10

Then the value of x will be,

→ 15 - x = 10

→ -x = 10 - 15

→ -x = -5

→ [ x = 5 ]

Hence, the value of x is 5.

Answer:

165/34

Step-by-step explanation:

245=80+(25+9)w We subtract 80 from each side and simplify

165=(34)w Divide each side by 34

165/34 = w

Answer:

9y^8

Step-by-step explanation:

15/5 = 3

y^6 / y^2 = y^4

so you are looking at basically (3y^4)² = 3y^4 * 3y^4 = 9y^8

If the reaction was done with excess sodium bicarbonate, the amount of calcium chloride would determine the amount of precipitate produced.

The theoretical mass of precipitate produced in a reaction between calcium chloride and sodium bicarbonate can be calculated by using the balanced chemical equation for the reaction. The balanced equation for this reaction is given as follows:

CaCl₂ + NaHCO₃ → CaCO₃ + NaCl + H₂O

In this reaction, one mole of calcium chloride reacts with one mole of sodium bicarbonate to produce one mole of calcium carbonate precipitate, one mole of sodium chloride solution, and one mole of water.

The mass of the precipitate can be calculated by multiplying the molar mass of calcium carbonate by the number of moles of calcium carbonate produced.

So, to calculate the theoretical mass of precipitate produced, you need to know the amount (in moles) of calcium chloride and sodium bicarbonate used in the reaction. If the reaction was done with excess sodium bicarbonate, the amount of calcium chloride would determine the amount of precipitate produced.

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Answer:

x = p(m-n)

Step-by-step explanation:

we can eliminate the 'p' in the denominator by multiplying each side of the equation by 'p' to get:

mp = np + x

now we can subtract 'np' from each side to get:

mp - np = x

we can further simplify it as:

p(m-n) = x

Answer:

X=n-p×m

Step-by-step explanation:

m=n-x/p

p×m=n-x

n-p×m=n+x-n

X=n-p×m

Answer:

10,37

Step-by-step explanation:

for EF= [tex]\sqrt{17^{2} -14^{2} } =10[/tex]

for D= we have 2 knowns which are 53 and 90, so we need to minus them from each other to get D.  90-53=37

The length of the rectangular prism is 4/5x.

What is a rectangular prism?

Six faces make up the three-dimensional shape of a rectangular prism (two at the top and bottom and four are lateral faces). The prism has rectangular shapes on each of its faces. There are therefore three sets of identical faces in this picture. A cuboid is another name for a rectangular prism because of its shape.

We know that Volume of a rectangular prism is a product of length, height and width.

We are given volume as (7/x)-(3/5x).

The width(w) is 2/x and the height(h) is 4/x

Using the formula, we get

⇒ V = lwh

⇒ (7/x)-(3/5x) = l * (2/x) * (4/x)

⇒ l = [(7/x)-(3/5x)] / [(2/x) * (4/x)]

⇒ l = (32/5x) /(8/x^2)

⇒ l = 4/5x

Hence, the length of the rectangular prism is 4/5x.

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The value of 21•15 equals (20+1)•15. This is true.

An expression is simply used to show the relationship between the variables that are provided or the data given regarding an information. In this case, it is vital to note that they have at least two terms which have to be related by through an operator. Some of the mathematical operations that are illustrated in this case include addition, subtraction, etc.

The value of 21•15 will be:

= 21 × 15

= 315

Also (20+1)•15 will be:

= 21 × 15

= 315

Therefore, they're equal.

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Complete question

The value of 21•15 equals (20+1)•15. True or false?

The required number in which when subtracted from both 17 and 12 , the product will be 1.04  is  x = 11.8 or 17.2.

ax^2 + bx + c = 0 is a quadratic equation, which is a second-order polynomial equation in a single variable. a 0. It has at least one solution because it is a second-order polynomial equation, which is guaranteed by the algebraic fundamental theorem.

Let the number is x, then

(17 - x)(12 - x) = 1.04

204 - 17x - 12x + x² = 1.04

x² - 29x + 204 = 1.04

x² - 29x + 202.96 = 0

On solving for x, we get

x = 11.8 or 17.2

Thus, the number can be x = 11.8 or 17.2.

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The needed value is either x = 11.8 or 17.2, which when deducted from both 17 and 12 will result in a product of 1.04.

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The number of years that will take for the schools to have the same number of students is given as follows:

11 years.

The number of students in each school is modeled by linear functions in slope-intercept format, as follows:

y = mx + b.

In which:

The functions for this problem are given as follows:

They will have the same number of students when:

Sp(t) = Se(t).

Hence:

1074 + 19t = 1206 + 7t

12t = 132

t = 132/12

t = 11 years.

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let f = father's present age, let s = son's

:

write an equation for each statement, simplify

:

Three years ago a father was thrice as old as his son

f - 3 = 3(s-3)

f - 3 = 3s - 9

f = 3s - 9 + 3

f = 3s - 6

in 5 years time the father will be twice as his son

f + 5 = 2(s+5)

f + 5 = 2s + 10

f = 2s + 10 - 5

f = 2s + 5

replace f with 3s - 6, from the 1st statement

3s - 6 = 2s + 5

3s - 2s = 5 + 6

s = 11 yrs is the son's age

find the fathers age

f = 3(11) - 6

f = 33 - 6

f = 27 yrs is the father's age

:

what will be the sum of their age in four years

27+4 + 11+4 = 46

Answer:

Step-by-step explain

You have to solve for 5a

The 14th term of the arithmetic sequence is 109 whose common difference is d = 8 and whose first term is a(1)=5.

The value of 14th term of an AP is 109.

In the given question

The common difference = 8

The value of first term=5

And we have to find the value of 14th term of Arithmetic progression

We utilize the method for locating the nth term to locate a specific term in an arithmetic series. An arithmetic sequence's nth term is determined by an = a + (n - 1)d in step 1.

Putting the given value in formula:

an = a + (n - 1)d

a 14th=5+(14-1)8

a 14th=5+13*8

a 14th=5+104

a 14th=109

So the value of 14th term of an AP is 109.

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The logarithmic function that describes the graph is given as follows:

[tex]f(x) = \log_2{(x + 1)}[/tex]

The parent logarithmic function is defined as follows:

f(x) = log(x).

Which, no matter the base, has the vertical asymptote given as follows:

x = 0.

The vertical asymptote of the graphed function is given as follows:

x = -1.

Hence:

[tex]f(x) = \log_a{(x + 1)}[/tex]

When x = 3, f(x) = 2, hence the base a is obtained as follows:

[tex]2 = \log_a{(4)}[/tex]

Meaning that:

a² = 4.

a = 2.

And the function is then defined as follows:

[tex]f(x) = \log_2{(x + 1)}[/tex]

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Let A be the point (-2, 9, 10) and n be the normal vector of the plane. The direction vector of the line is <1, 3, -3>, and since the plane is perpendicular to the line, n is orthogonal to the direction vector.

So, n is a vector that satisfies:

n . <1, 3, -3> = 0

The equation of the plane can be written in the form:

n . (x - A) = 0

Where x = (x, y, z) is any point in the plane.

We can find n by taking the cross product of two non-parallel vectors in the plane, such as <1, 0, 0> and <0, 1, 0>. Then,

n = (1, 0, 0) x (0, 1, 0) = (0, 0, 1)

Substituting the values of A and n into the equation of the plane, we get:

0 . (x - (-2, 9, 10)) = 0

0 . (x + 2, y - 9, z - 10) = 0

z - 10 = 0

z = 10

So the equation of the plane is:

z = 10

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The function that can be used to get the height of the helicopter above the ground is: H=280m−1,500

A function is a statement, rule, or law in mathematics that establishes the link between an independent variable and a dependent variable (the dependent variable).

How to solve for the function

The value 280 is the constant rate that the helicopter is said to be descending.

The altitude from which it is coming down is given as 1500 feet

The function would be written as H=280m−1,500

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The relation between the number of hours spent using electronic devices is 2 times the number of hours spent playing sports. ​

The term called relation is defined as the relationship between sets of values of ordered pairs and the set of elements in the first set are called domain which is related to the set of the element in another set, which is called range.

Here we have know that the number of hours spent using electronic devices is 30 and the number of hours spent playing sports 15.

As per the definition of relation, in order to find the relation we need to do the division operation on then we get,

=> 30/15

=> 2

Which means that the first one is 2 times of the second one.

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Answer:

a) The two variables are the number of years and the height of the tree in feet

b) It is a positive linear relationship.

c) Put a point at (0,3) and one at (3,7), and connect the dots.

Answer:

1. y= (x-3)^2+5 vertex form

2. y= 2(x+2)^2+4 vertex form

3. y= 3x^2+18x+20 general form

4. y= x^2+4x+8 general form