Arjun is building a frame for his father's painting. Which measurement will tell him the minimum amount of wood that he
will need to build the frame?

A.the perimeter of the painting

B.the volume of the painting

C.the width of the painting

D.the area of the painting

In the given scenarios: 1. Car speed (x) determines distance covered (y) in 30 minutes (y = 30x). 2. Tina's age (y) is 5 years more than Tim's age (x) (y = x + 5). 3. Chair cost (x) relates to table cost (y) as y = 2x. 4. Baking time for chocolate muffins (y) is 2 minutes longer than for vanilla muffins (x) (y = x + 2). These equations establish connections between variables in different real-world situations.

In these scenarios, we encounter different situations where certain variables and their relationships are described. Let's elaborate on each scenario and express it in a more detailed manner:

1. **Car Speed and Distance:** The speed of a car is represented by the variable 'x' in miles per hour. The total distance covered by the car in 30 minutes, denoted as 'y,' can be expressed using the formula: y = 30x. This equation relates the speed 'x' to the distance 'y' covered in a specific time period.

2. **Sibling Ages:** Tim's age, represented by 'x,' and Tina's age, represented by 'y,' are related. Tina's age is 5 years more than Tim's age, which can be expressed as y = x + 5. This equation defines the age difference between the siblings.

3. **Cost of Furniture:** In this scenario, the cost to produce a chair is 'x,' and the cost to produce a table is 'y.' The relationship is that the cost of producing a table is two times the cost of producing a chair, which can be expressed as y = 2x. This equation shows the cost relationship between chairs and tables.

4. **Muffin Baking Times:** The time required to bake a vanilla muffin is 'x,' and the time required to bake a chocolate muffin is 'y.' The relationship is that the time to bake a chocolate muffin is 2 minutes more than the time to bake a vanilla muffin, which can be expressed as y = x + 2. This equation establishes the time difference between baking these two types of muffins.

In summary, these scenarios involve different relationships and equations that connect variables and quantities in various real-world contexts, such as car speed and distance, sibling ages, furniture production costs, and muffin baking times. These equations help describe and quantify these relationships, making them useful for solving practical problems.

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Answer:

see explanation

Step-by-step explanation:

Given

[tex]log_{10}[/tex] x = 0.98, then

x = [tex]10^{0.98}[/tex]

The value of (5.3x 10^4) (4.2x10^3) in scientific notation Option C 2.226 times 10^8

5.3×10^4  × 4.2 × 10^3

We multiply the numbers out front

5.3× 4.2 = 22.26

We add the exponents

10^(4+3) = 10^7

22.26×10^7

But we are not done yet because The number out front has to be between 1 and less than 10

Move the decimal 1 place to the left and add 1 to the exponent

2.226 × 10 ^(7+1)

=2.226×10^8.

Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10^8.

Scientific Notation is the expression of a number n in the form a∗10b. where a is an integer such that 1≤|a|<10. and b is an integer too. Multiplication: To multiply numbers in scientific notation, multiply the decimal numbers.

The proper format for scientific notation is a x 10^b where a is a number or decimal number such that the absolute value of a is greater than or equal to one and less than ten or, 1 ≤ |a| < 10. b is the power of 10 required so that the scientific notation is mathematically equivalent to the original number.

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Answer:

option A.

y = (1/2)x - 2

Step-by-step explanation:

It is given that, line that passes through the points (2, –1) and (6, 1)

Slope m = (y₂ - y₁)/(x₂ - x₁)

 = (1 - -1)/(6 - 2)

 = (1 + 1)/4 = 2/4

 = 1/2

To find the equation of the line

Equation of a line passing through the points (x₁, y₁)  and  slope  m is given by,

(y - y₁)/(x - x) = m

Here (x₁, y₁) = (2, -1) and m = 1/2

(y - -1)/(x - 2) = 1/2

(y + 1) = (x - 2)/2

y = -1 + x/2 - 2/2

  =  -1 + x/2 - 1

 = x/2 - 2

The equation of the line is,

y = (1/2)x - 2

The correct answer is option A.

y = (1/2)x - 2

Answer:

C

Step-by-step explanation:

Each term in the second factor is multiplied by each term in the first factor, that is

- 2x(- 4x - 3) - 9y²(- 4x - 3) ← distribute both parenthesis

= 8x² + 6x + 36xy² + 27y² → C

Answer:

The correct answer is option D.  85°

Step-by-step explanation:

Here we consider two angles be <1, <2 and < 3, where <1 is the linear pair of angle measures 133° and <2 be the linear pair of angle measures 142°

To find the value of m<1

m<1 = 180 - 133 = 47°

To find the value of m<2

m<2 = 180 - 142 = 38°

To find the value of m<3

By using angle sum property,

m<1 + m<2 + m< 3 = 180  

m<3 =180 - (m<1 + m<2)

 = 180 - (47 + 38))

 = 180 - 85

 = 95°

To find the value of n

Here n and <3 are linear pair,

n + m<3 = 180

n = 180 - m<3

 = 180 - 95 = 85°

Therefore the value of n = 85°

The correct answer is option D.  85°

Answer:

The page numbers are 36 and 37.

Step-by-step explanation:

n + (n + 1) = 73

2n + 1 = 73

n = (73 - 1)/2

= 72/2 = 36

Follow below steps:

The sum of the page numbers on the facing pages of a book is 73. To solve this, we need to establish that facing pages in a book are always one number apart. If we call one page number x, then the other page number is x+1 because page numbers are consecutive.

Now, we can create the equation x + (x+1) = 73 to find the value of x. Solving this equation, we get:

2x + 1 = 73

2x = 73 - 1

2x = 72

x = 72 / 2

x = 36

Since x is the lower of the two page numbers, then the pages are 36 and 37.

Answer:

B. y + 2 = ½(x + 3)

Step-by-step explanation:

Insert the coordinates into the formula with their CORRECT signs. Remember, in the Point-Slope Formula, y - y₁ = m(x - x₁), all the negative symbols give the OPPOSITE term of what they really are.

Answer:

10x + 4

Step-by-step explanation:

The Total perimeter is 2 * leg measurement + base

P = 2(2x + 3) + 6x - 2           Remove the brackets.

P = 4x + 6 + 6x - 2               Add like terms

P = 10x + 4                           Answer

Answer:

10x + 4

Step-by-step explanation:

An isosceles triangle has two sides equal

Perimeter is the sum of all sides

2(2x + 3) + 6x - 2

4x + 6 + 6x - 2

10x + 4

Answer: Inconsistent.

Step-by-step explanation:

The  equation of the line in Slope-Intercept form is:

[tex]y=mx+b[/tex]

Where "m" is the slope and "b" is the y-intercept.

Solve for "y" in each equation:

Equation 1

[tex] x=-5-y\\\\y=-x-5[/tex]

Equation 2

[tex]4+y=-x+3\\\\y=-x+3-4\\\\y=-x-1[/tex]

You can notice that the slope of the Equation 1 is:

[tex]m_1=-1[/tex]

And the slope of the Equation 2 is:

[tex]m_2=-1[/tex]

Observe that [tex]m_1=m_2[/tex], then you can conclude that the lines are parallel and the System of equations has No solution.

When there is no solution the classification  of the system of equations is: "Inconsistent".

Answer:

[tex]\large\boxed{\dfrac{V_{cone}}{V_{prism}}=\dfrac{1}{3}}[/tex]

Step-by-step explanation:

[tex]\text{The formula of a volume of a cone:}\ V_{cone}=\dfrac{1}{3}B_cH_c\\\\B_c-base\ area\ of\ a\ cone\\H_c-height\ of\ a\ cone\\\\\text{The formula of a volume of a prism:}\ V_{prism}=B_pH_p\\\\B_p-base\ area\ of\ a\ prism\\H_p-height\ of\ a\ prism\\\\\text{The cone and the prism have the same base area and height.}\\\text{Therefore}\\\\V_{cone}=\dfrac{1}{3}BH\ \text{and}\ V_{prism}=BH\\\\\text{The ratio of the volume of the cone to the volume of the prism:}[/tex]

[tex]\dfrac{V_{cone}}{V_{prism}}=\dfrac{\frac{1}{3}BH}{BH}=\dfrac{1}{3}[/tex]

Answer:

See the image attached for answer

Step-by-step explanation:

a1+a2+a3=21

(a1=1,a2=3,a3=7)

a1xa2xa3=64

(a1=2,a2=4,a3=8)

Perimeter: 2W +2L = 48

Area = L x W = 140

Rewrite the area to solve for L: L = 140/W

Now replace that in the perimeter formula:

2W + 2(140/W) = 48

Divide all terms by 2:

W + 140/W = 24

Divide both sides by W:

140 + w^2 = 24w

Subtract 24w from both sides:

w^2 - 24w + 140 = 0

Factor:

(w-10) (w-14) = 0

Solve for each w for 0:

10-10 - 10 and 14-14 = 0

So the 2 dimensions are 10 and 14 cm.

14 isn't a choice so the answer is B. 10 cm.

Answer:

8 inches

Step-by-step explanation:

To find the area of a triangle, we use the formula

A = 1/2 bh where b is the length of the base and h is the height

A = 24 and b = 6

24 = 1/2 (6) * h

24 =3h

Divide each side by 3

24/3 = 3h/3

8 =h

The height is 8 inches

The answer is:

The height of the triangle is equal to 8 inches.

[tex]height=8in[/tex]

To find the height of the triangle, we need to use the formula to calculate the area of a triangle, it's given by the following expression:

[tex]Area=\frac{base*height}{2}[/tex]

So, since we already know the area and the base, we need to isolate the height from the formula, so, isolating we have:

[tex]Area=\frac{base*height}{2}\\\\\frac{Area*2}{base}=height[/tex]

We know that:

[tex]Area=24in^{2} \\base=6in[/tex]

Now, substituting we have:

[tex]height=\frac{Area*2}{base}[/tex]

[tex]height=\frac{24in^{2} *2}{6in}=8in[/tex]

Hence, we have that the height of the triangle is equal to 8 inches.

[tex]height=8in[/tex]

Have a nice day!

Answer:

[tex]\large\boxed{(-2z^2-14)-(-12z^2)=10z^2-14}[/tex]

Step-by-step explanation:

[tex](-2z^2-14)-(-12z^2)=-2z^2-14+12z^2\qquad\text{combine like terms}\\\\=(-2z^2+12z^2)-14=10z^2-14[/tex]

Answer:

5:7

Step-by-step explanation:

Ratio of the areas of similar shapes is equal to the square of the scale:

A₂ / A₁ = r²

Ratio of the side lengths of similar shapes is equal to the scale:

s₂ / s₁ = r

Therefore:

A₂ / A₁ = (s₂ / s₁)²

25/49 = (s₂ / s₁)²

s₂ / s₁ = 5/7

The ratio of the side lengths is 5:7.

Answer:

5 : 7

Step-by-step explanation:

Given 2 similar figures then

ratio of sides = a : b

ratio of areas = a² : b²

Here the ratio of areas = 25 : 49, hence

ratio of sides = [tex]\sqrt{25}[/tex] : [tex]\sqrt{49}[/tex] = 5 : 7

The expression a/b is evaluated by dividing -6 by -2. Since both numbers are negative, the result is a positive 3.

The problem asks you to evaluate a/b with a=-6 and b=-2. The expression a/b means 'a divided by b'. Thus, to evaluate this expression, you simply divide -6 by -2. Remember that when you divide a negative number by another negative number, the result is positive. So, -6 divided by -2 equals 3.

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We substitute -6 for a, and -2 for b in the expression a/b, obtaining -6/-2, which equals 3 because division of two negative numbers yields a positive number.

To evaluate the expression a/b for a = -6 and b = -2, we can substitute the given values into the expression. This gives us -6 / -2. Dividing a negative number by another negative number gives a positive result. Therefore, -6 / -2 equals 3. The notion that dividing a negative number by another negative number results in a positive number is a fundamental rule in mathematics.

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Answer:

20.01

Step-by-step explanation:

The mean of 18, 24 and 18.05 is 20.01.

Step-by-step explanation:

solved in the picture....

Answer:

The factored form of the given expression is [tex]3x(x^2+4x+6)[/tex].

Step-by-step explanation:

The given expression is

[tex]3x^3+12x^2+18x[/tex]

we need to find the factored form of the given expression.

Taking GCF common, we get

[tex]3x(x^2+4x+6)[/tex]

If a quadratic expression is defined as [tex]ax^2+bx+c[/tex] and

[tex]b^2-4ac<0[/tex], then the expression can not be factored further.

In the above parentheses the quadratic expression is [tex]x^2+4x+6[/tex],

[tex]b^2-4ac=(4)^2-4(1)(6)=-8<0[/tex]

It means the quadratic expression is [tex]x^2+4x+6[/tex] can not be factored further.

Therefore the factored form of the given expression is [tex]3x(x^2+4x+6)[/tex].

Answer: 203.31

Step-by-step explanation:

Each bus have 214.42.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value

Give:

Total buses=9

Charges for tip= $ 100

Total money spends = $ 1829.82

Amount with tip = $ 1829.82+$ 100 = $ 1929.82

So, each bus have = 1929.82/9

                              = 214.42

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Answer:

12.75 oz

Step-by-step explanation:

1 gallon is equivalent to 64 oz.

Writing and solving an equation of ratios, we get:

 1.5 oz       1 gal

----------- = ------------

     x          8.5 gal

Here we must cross-multiply to solve for x.

x(1 gal) = (1.5 oz)(8.5 gal) = 12.75 oz

Thus, x = 12.75 oz of cleaning product must be added to 8.5 gal of water to create the desired solution.

To make an all-purpose cleaner, you need 12.75 ounces of cleaner for 8.5 gallons of water.

This is determined by multiplying the ratio of 1.5 ounces per gallon by 8.5 gallons.

follow these steps:

Therefore, 12.75 ounces of cleaner will be added to 8.5 gallons of water.

Answer:

see explanation

Step-by-step explanation:

Under a reflection in the x- axis

a point (x, y ) → (x, - y ), hence

F(- 1, 9) → F'(- 1, - 9)

G(- 2, 1) → G'(- 2, - 1)

H(- 7, 4) → H'(- 7, - 4)

Answer:

0.75 so FALSE

Step-by-step explanation:

It is the only number that could be left after 0.5 and 0.25

Answer:

y=x-4

Step-by-step explanation:

Standard form is y=mx+b

the coordinates (0, -4) tell you that the y intercept is at -4 which means b will be -4. To find slope you have to find the change in y over the change is x. 1--4=5 and 5-0=5. 5/5 is 1 so the slope, or m, is 1. x stays the same since it's the variable. it'll become y=1x-4 and since you don't have to place a coordinate of one in front of a variable, you can just write it as y=x-4

Answer:

(0.1,1.1)

Step-by-step explanation:

we have

[tex]f(x)=3^{x}[/tex]

[tex]g(x)=-log(x)[/tex]

Solve the system of equations by graphing

The solution is the intersection point both graphs

The solution is the point (0.081,1.093)

see the attached figure

Round to the nearest tenth ----> (0.1,1.1)

Answer:

Step-by-step explanation:

The intercept form of a quadratic equation y = ax² + bx + c:

[tex]y=a(x-p)(x-q)[/tex]

[tex]\text{x-intercepts:}\ p\ \text{and}\ q\\\\\text{y-intercept}:\ a(-p)(-q)[/tex]

We have the equation:

[tex]y=2x^2-9x-5=(2x+1)(x-5)[/tex]

[tex]2x+1=2\left(x+\dfrac{1}{2}\right)\to y=2\left(x+\dfrac{1}{2}\right)(x-5)[/tex]

[tex]y=2\bigg(x-\left(-\dfrac{1}{2}\right)\bigg)(x-5)[/tex]

Therefore

[tex]a=2\\\\x-intercepts:\ p=-\dfrac{1}{2}\ \text{and}\ q=5\\\\\text{y-intercept:}\ (2)\left(-\dfrac{1}{2}\right)(5)=-5[/tex]

The x-intercepts are -1/2 and 5, and the y-intercept is -5.

The factored form of a quadratic equation is y=(2x+1)(x-5). To find the x-intercepts, we set y=0 and solve for x. Therefore:

The y-intercept is found by setting x=0 and solving for y. So:

Therefore, the correct statement is:

The x-intercepts are -1/2 and 5, and the y-intercept is -5.

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Answer:

D. 96 inches

Step-by-step explanation:

Perimeter (P) = 2L + 2W , Where L is the length and W is the width.

20 ft = 2L + 2(2 ft)

2L = 20 ft - 4 ft = 16 ft

L = [tex]\frac{16}{2}[/tex] ft = 8 ft

1 ft = 12 inches

8 ft = ?

Cross-multiplying this gives; [tex]\frac{8}{1}[/tex] × 12 = 96 inches

Answer: D) 96 inches.

Step-by-step explanation: To calculate the lenght of James' rectangular yard, we need to isolate L from the given formula:

P=2L+2W

P-2W=2L

[tex]L=\frac{P-2W}{2}[/tex]

Now we replace the given values of P and W in the equation:

[tex]L=\frac{20feet-2*2feet}{2}[/tex]

[tex]L=\frac{20feet-4feet}{2}[/tex]

[tex]L=\frac{16feet}{2}[/tex]

[tex]L=8feet[/tex]

1feet=12inches so:

L=8feet*12inches/1feet

L=96 inches.

Answer:

0.0005 oz

Step-by-step explanation:

Usually, the greatest number that is allowed for approximation, assuming that the number itself is obtained by approximation, is the greatest possible error of it.

It is normally half the place value of the last digit in a number.

Like here we have [tex]10\frac{1}{8}[/tex] oz which is equal to [tex]10.125[/tex] oz. The last digit is 5 which is at the thousandth place (0.001) so the greatest possible error for this would be its half.

[tex]\frac{0.001}{2}[/tex] = 0.0005 oz

To estimate the value of 98.5 x 13, round the final answer to the hundredths position. In this case, round up to 922.00.

To estimate the value of 98.5 x 13, we can use the rounding rule that states if the first digit to be dropped is greater than 5, we round up. In this case, the digit in the thousandths place is greater than 5, so we round up to the hundredths position. Therefore, the estimated value is 922.00.

Answer:

The radius of the base of the cone is 2 units

The slant height of the cone is 4 units

The height of the cone is 2√3 units

The volume of the cone is [tex]\frac{8\sqrt{3}\pi}{3}[/tex]units³

Step-by-step explanation:

* Lets revise the total surface area and the lateral area of a cone

- The lateral area of cone = π r l , where r is the radius of the base

  and l is the slant height of the cone

- The surface area of the cone = π r l + π r², where π r l is the lateral

  area and π r² is the base area

- The cone has three dimensions radius (r) , height (h) , slant height (l)

- r , h , l formed right triangle, r , h are its legs and l is its hypotenuse,

 then l² = r² + h²

- The volume of the con = [tex]\frac{1}{3}[/tex] (π r² h)

* Now lets solve the problem

- We will use the total area to find the radius of the base

∵ TA = 12π

∵ TA = LA + πr²

∵ LA = 8π

- Substitute the value of the lateral area in the total area

∴ 12π = 8π + π r² ⇒ subtract 8π from both sides

∴ 12π - 8π = π r²

∴ 4π = π r² ⇒ divide both sides by π

∴ r² = 4 ⇒ take square root for both sides

∴ r = 2

* The radius of the base of the cone is 2 units

- We will use the lateral area to find the slant height

∵ LA = π r l

∵ LA = 8π

∵ r = 2

∴ π (2) l = 8π ⇒ divide both sides by π

∴ 2 l = 8 ⇒ divide both sides by 2

∴ l = 4

* The slant height of the cone is 4 units

- Use the rule l² = r² + h² to find the height of the cone

∵ r = 2 and l = 4

∵ l² = r² + h²

∴ (4)² = (2)² + h²

∴ 16 = 4 + h² ⇒ subtract 4 from both sides

∴ 12 = h² ⇒ take square root for both sides

∴ h = √12 = 2√3

* The height of the cone is 2√3 units

∵ The volume of the con = [tex]\frac{1}{3}[/tex] (π r² h)

∵ r = 2 and h = 2√3

∴ V = [tex]\frac{1}{3}[/tex] (π × 2² × 2√3) = [tex]\frac{1}{3}[/tex] (π × 4 × 2√3) = [tex]\frac{1}{3}[/tex] (π × 8√3)

∴ V = [tex]\frac{8\sqrt{3}\pi}{3}[/tex]

* The volume of the cone is [tex]\frac{8\sqrt{3}\pi}{3}[/tex]units³