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The length of the side of the base is 6 units.

The volume of the pyramid is;

[tex]\rm Volume \ of \ pyramid=\dfrac{1}{3}\times height \times Base[/tex]

The pyramid shown below has a square base, a height of 7, and a volume of 84.

Substitute all the values in the formula

[tex]\rm Volume \ of \ pyramid=\dfrac{1}{3}\times height \times Base\\\\\rm84 =\dfrac{1}{3}\times height \times 7\\\\Height = \dfrac{84\times 3}{7}\\\\ Height = \dfrac{252}{7}\\\\Height =36[/tex]

The length of the side of the base is;

[tex]\rm Side\ length^2=height\\\\ Side\ length^2=36\\\\Side\ length^2=6^2\\\\Side\ length=6[/tex]

Hence, the length of the side of the base is 6 units.

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To solve the given quadratic equation, we simplified and factored it, yielding solutions x = 0 and x = 17. For the data in the table, an exponential function is suggested by the consistent ratio between y-values. Algebraically solving the system of equations results in two solutions: (5, 35) and (-4, 8).

Formula to Solve the Equation

To solve the equation x squared minus 21 x equals negative 4 x, you'll first want to rewrite it in standard form. Moving all the terms to one side gives us [tex]x^2 - 21x + 4x = 0[/tex], which simplifies to [tex]x^2 - 17x = 0[/tex]. To find the solutions, we can factor out an x, giving us x(x - 17) = 0. This yields two solutions: x = 0 and x = 17.

Model the Data in the Table

Looking at the data given, we see that each y-value seems to be a multiple of the previous y-value. This suggests an exponential function might be a better fit than a linear one. To confirm, you can divide each y-value by the previous y-value and observe if the ratios are consistent. Calculating the ratios between consecutive y-values in the table, we see that the ratio is consistently around 4, confirming that an exponential model is appropriate.

Solve the System of Equations Algebraically

To solve the system [tex]y = x^2 + 2x[/tex]and y = 3x + 20, set the right sides of the equations equal to each other:[tex]x^2 + 2x = 3x + 20[/tex]. Bringing all terms to one side gives us the quadratic equation[tex]x^2 - x - 20 = 0[/tex]. Factoring the quadratic equation we get (x - 5)(x + 4) = 0, which gives us two solutions for x: x = 5 or x = -4. Substituting these back into either original equation provides the related y-values, thus giving us two solutions for the system: (5, 35) and (-4, 8).

Answer:

Because in one way or another, you would lose money.

Step-by-step explanation:

Sometimes, it is a good idea to underprice a house when selling it, specially when there is a hot market to start a bidding war. If this is not the case, when the prices go down, it means that the seller must adjust his prices to the market; meaning that the lower the prices are, the lower the underpricing should be, so the seller would lose money.

The estimated amount of fabric that the umbrella has is approximately [tex]\(112 + 32\sqrt{2}\)[/tex] square feet.

To estimate the amount of fabric that the umbrella has, we first need to find the total surface area of the regular octagonal pyramid.

A regular octagonal pyramid consists of 8 congruent isosceles triangles as its lateral faces and a regular octagon as its base.

Given:

- Side length of the octagon (s) = 4 feet

- Slant height of the pyramid (l) = 5 feet

First, let's calculate the area of one of the lateral faces (isosceles triangle) using the formula for the area of a triangle:

[tex]\[ \text{Area of one lateral face} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]

In this case, the base of the triangle is the side of the octagon, s=4 feet, and the height is the slant height of the pyramid, l=5 feet. Therefore:

[tex]\[ \text{Area of one lateral face} = \frac{1}{2} \times 4 \times 5 = 10 \text{ square feet} \][/tex]

Since there are 8 congruent lateral faces on the octagonal pyramid, the total area of the lateral faces is:

[tex]\[ \text{Total area of lateral faces} = 8 \times 10 = 80 \text{ square feet} \][/tex]

Now, let's calculate the area of the base (regular octagon). The formula for the area of a regular octagon is:

[tex]\[ \text{Area of octagon} = 2 \times (1 + \sqrt{2}) \times s^2 \]\[ \text{Area of octagon} = 2 \times (1 + \sqrt{2}) \times 4^2 \]\[ \text{Area of octagon} \approx 2 \times (1 + \sqrt{2}) \times 16 \]\[ \text{Area of octagon} \approx 32 \times (1 + \sqrt{2}) \][/tex]

Now, we can calculate the total surface area of the regular octagonal pyramid by adding the area of the lateral faces and the area of the base:

[tex]\[ \text{Total surface area} = \text{Area of lateral faces} + \text{Area of octagon} \]\[ \text{Total surface area} = 80 + 32 \times (1 + \sqrt{2}) \]\[ \text{Total surface area} \approx 80 + 32 \times (1 + \sqrt{2}) \]\[ \text{Total surface area} \approx 80 + 32 + 32\sqrt{2} \]\[ \text{Total surface area} \approx 112 + 32\sqrt{2} \][/tex]

Subtract 3 from each side

y - 3 = 2x

Divide through by 2

y/2 - 3/2 = x

or x = y/2 - 3/2

Now simply switch labels: x↔y

y = x/2 - 3/2

The elimination method is used for solving linear systems of equations.It is way of solving a linear system of equations.In the elimination method we either add or subtract the equations to get an equation in one variable.The main concept behind elimination method is to create terms with opposite coefficients because they cancel each other when added.

When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

When Heather has to solve this system of equations using elimination her first step will be to make the coefficients of x or y the same   so that one of the variable is eliminated on addition or subtraction of the equations.

Answer: 34 + y  = 97

Step-by-step explanation:

According to the question,

Total number of sport cards = 97

In which some are foot ball cards and some are non football cards.

Let y be the number of sports cards.

Given, 34 cards are football cards.

Therefore, total number of cards = 34 + y

⇒ 34 + y = 97

Which is the equation that will use to find the number of non football cards.

To find the number of sports cards that are not football cards, you subtract the number of football cards from the total number of sports cards. The equation is y = 97 - 34.

To determine the number of sports cards that are not football cards, you need to subtract the number of football cards from the total number of sports cards.

Step-by-Step Explanation:

The total number of sports cards is 97.

The number of football cards is 34.

The number of sports cards that are not football cards can be found by subtracting the number of football cards from the total number of sports cards: y = 97 - 34.

Therefore, the equation y = 97 - 34 can be used to find the number of sports cards that are not football cards.

The solution is, 308π in² is the surface area of the cylinder in terms of pi.

Surface area is the amount of space covering the outside of a three-dimensional shape. The surface area of a solid object is a measure of the total area that the surface of the object occupies.

here, we have,

from the given diagram , we get,

the cylinder has,

radius = r = 7

height = h = 15

we know that,

Surface Area = 2πr² + 2πrh

so, we get,

r = 7

h = 15

substituting the values, we get,

Surface Area

= 2π(7)² + 2π(7)(15)

= 308π in²²

Answer: 308π in²

Hence, The solution is, 308π in² is the surface area of the cylinder in terms of pi.

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The probability of selecting correct options is 1/3.

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur. Mathematically -

P[E] = n[E]/n[S]

Given are first 2 questions on a multiple choice test such that each of the question has 3 possible answers.

Now, there are two questions, with three possible answers, this means that we have total of 6 elements in a sample space. Let [A] represent the event of selecting correct options. Then there could be only 2 possible answers from the given six. Therefore, the probability of the event [A] would be :

P[A] = n[A]/n[S]

P[A] = 2/6

P[A] = 1/3

Therefore, the probability of selecting correct options is 1/3.

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[tex]\huge \boxed{\sf 26\sqrt{2}\ cm } \\\\\\\sf Using\ trigonometric\ functions \\\\\displaystyle cos \theta = \frac{adj}{hyp} \\\\cos(60)=\frac{adj}{52} \\\\adj=cos(60) \times 52=26 \\\\Using\ the\ Pythagorean\ theorem \\\\x=\sqrt{26^2+26^2} =26\sqrt{2}[/tex]

Final answer:

To find out if the ratios 1/9 and 6/54 form a proportion, calculate their cross-products. Since both cross-products are equal (54 = 54), they do form a proportion.

Explanation:

The question asks if two given ratios form a proportion. To determine whether ratios form a proportion, you can compare the cross-products of the ratios. If the cross-products are equal, the ratios do form a proportion. For the given ratios 1/9 and 6/54, we will calculate the cross-products:

For the first ratio, 1 times 54 equals 54.

For the second ratio, 9 times 6 equals 54.

Since both cross-products are equal (54 = 54), the ratios 1/9 and 6/54 do indeed form a proportion.

To write a proportion by setting two ratios equal to one another with the unit 'meters', an example would be 1/20 = 1/5.5. When working with unit scales or unit rates, such as in map readings or speed, you compare two measurements where one of the ratios is typically a value of 1, like the example provided 55 miles per hour, which is written as 55/1 miles/hour.

Answer:

205 cubic units

Step-by-step explanation:

Given : The expression 100 + 20m gives the volume of water in Eduardo’s pool (in liters) after Eduardo spends m minutes filling his pool.

To Find: What is the volume of water in Eduardo’s pool after he fills it for 5.25 minutes?

Solution:

Expression: Volume of water in pool after m minutes= 100 + 20m

Volume of water in pool after m minutes = 100 + 20m

Where m is the number of minutes spent in filling pool.

Now we are supposed to find the volume of water in Eduardo’s pool after he fills it for 5.25 minutes

Substitute m = 5.25

Volume of water in pool after m minutes= 100 + 20(5.25)

                                                                  =205

Thus the volume of water in Eduardo’s pool after he fills it for 5.25 minutes is 205 cubic units.

Step-by-step explanation:

[tex]cos \: x = \frac{a}{h} \\ cos \: x = \frac{16}{21} = 0.76 \\ cos \: x = 40.36degrees[/tex]

The total cost of purchase on 2 pants and 1 shirt is $76.986.

Given that, you bought 2 pairs of pants at $28.99 each, and one shirt at $21.34.

Calculating the sales tax applied to a purchase is a matter of simply multiplying the tax rate by the purchase price using the equation sales tax = purchase price × sales tax rate. Adding the sales tax to the original purchase price gives the total price paid with tax.

Now,

The cost of 2 pairs of pants

= 2 ×28.99

= $57.98

Total cost of two pants and a shirt

= 21.34+51.98

= $73.32

Tax =5% of 73.32

= 5/100 ×73.32

= 0.05 ×73.32

= $3.666

Total cost of purchase =73.32+3.666

= $76.986

Therefore, the total cost of purchase on 2 pants and 1 shirt is $76.986.

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