The St. Louis Cardinals play 160 games in a season. So far, they have won 50 games. How many more games must they win in order to win at least 68% of all games for the season?

Answer:

D and E

Step-by-step explanation:

Using the rule of radicals

[tex]a^{\frac{m}{n} }[/tex] ⇔ [tex]\sqrt[n]{a^{m} }[/tex]

Given

[tex]3^{\frac{4}{7} }[/tex]

= ([tex]\sqrt[7]{3}[/tex])^4 → D or

= [tex]\sqrt[7]{3^{4} }[/tex] = [tex]\sqrt[7]{81}[/tex] → E

Answer: H-7

Step-by-step explanation: I got it right on test!  have a great day! :)

Answer:

x=5, y=22. (5, 22).

Step-by-step explanation:

y-x=17

y=4x+2

----------

4x+2-x=17

3x+2=17

3x=17-2

3x=15

x=15/3

x=5

y-5=17

y=17+5

y=22

Answer:

Step-by-step explanation:

[tex]\left\{\begin{array}{ccc}y-x=17&(1)\\y=4x+2&(2)\end{array}\right\\\\\\\text{substitute (2) to (1):}\\\\(4x+2)-x=17\qquad\text{combine like terms}\\\\(4x-x)+2=17\qquad\text{subtract 2 from both isdes}\\\\3x+2-2=17-2\\\\3x=15\qquad\text{divide both sides by 3}\\\\\dfrac{3x}{3}=\dfrac{15}{3}\\\\x=5[/tex]

[tex]\text{Put the value of x to (2):}\\\\y=4(5)+2\\\\y=20+2\\\\y=22[/tex]

Solve: 4(6x - 10) = 8x + 40

A 0

B.5/2

c. 23

D. 5​

Option D

The solution to given equation is x = 5

Given that we have to solve the given equation

4(6x - 10) = 8x + 40

Let us solve the above expression and find value of "x"

Multiplying 4 with terms inside bracket in L.H.S we get,

24x - 40 = 8x + 40

Move the variables to one side and constant terms to other side

24x - 8x = 40 + 40

Combine the like terms,

16x = 80

[tex]x = \frac{80}{16} = 5[/tex]

Thus solution to given equation is x = 5

Final answer:

To find the value of d in the equation 749/d ∙ d/749 = 1, we can simplify the equation and solve for d. Canceling out the like terms and cross multiplying leads us to the solution: d = 1.

Explanation:

To find the value of d in the equation 749/d ∙ d/749 = 1, we need to simplify the equation and solve for d.

We can start by canceling out the like terms: 749 and d. Multiplying the numerators and denominators gives us:
(749/d) ∙ (d/749) = (749 ∙ d) / (d ∙ 749)

Since the two fractions are equal to 1, we can set up the equation:
(749 ∙ d) / (d ∙ 749) = 1

Next, we can cross multiply:
749 ∙ d = d ∙ 749

Now, we can divide both sides of the equation by 749:
d = d ∙ 749 / 749

Simplifying further:
d = 749 / 749

Finally, we arrive at the solution:
d = 1

Answer:

altitude = 10 cm

Step-by-step explanation:

Finding the altitude of triangle when area and base is given:

         [tex]\sf \boxed{\text{\bf Area of triangle = $\dfrac{1}{2}*base*altitude$}}[/tex]

                               base = 20 cm

          Area of a triangle = 100 sq.cm

               [tex]\sf \dfrac{1}{2}*20*altitude = 100\\\\\\[/tex]

                   10 * altitude = 100      

                          altitude = 100 ÷ 10

                         altitude = 10 cm

There were 45 markers originally in each box.

Step-by-step explanation:

Given,

Boxes ordered = 3 large boxes

Markers given to one teacher = 15

Markers given to 3 teachers = 15*3 = 45 markers

Remaining markers = 90

Let,

x be the original number of markers in 3 boxes.

Total markers - markers given to teachers = markers left

[tex]x-45=90\\x=90+45\\x=135[/tex]

There were 135 markers in 3 boxes.

3 boxes = 135 markers

1 box = [tex]\frac{135}{3}=45\ markers[/tex]

There were 45 markers originally in each box.

Keywords: multiplication, addition

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Answer:

Area of the path is 475 m².

Step-by-step explanation:

Let us first draw the diagram for the given question.

Here rectangle ABCD represents a park whose length is 55 m and breadth is 35 m. The shaded portion shows the path all around the park whose width is 2.5 m.

Now, length of rectangle ABCD, l = 55 m

breadth of rectangle ABCD, B = 35 m

Now, width of the path = 2.5 m

So, length of rectangle PQRS, L = 55 + 2.5 + 2.5 = 60 m

breadth of rectangle PQRS, B = 35 + 2.5 + 2.5 = 40 m

Now, to calculate the area of shaded portion or the area of the path, we will subtract the area of the park or area of rectangle ABCD from the area of rectangle PQRS.

Now, area of rectangle PQRS, A₁ = L × B = 60 × 40 = 2400 m²

Area of rectangle ABCD, A₂ = l × b = 55 × 35 = 1925 m²

So, area of the path = A₁ - A₂ = 2400 - 1925 = 475 m²

Hence the area of the path is 475 m².

Answer:

10. ○[tex]\displaystyle 4,85; 4\frac{17}{20}[/tex]

9. ○680%

8. ○40%

7. ○[tex]\displaystyle 60[/tex]

6. ○0,4, 40,5%, 11⁄25, 4⁄9

5. ○[tex]\displaystyle 0,928[/tex]

4. ○1%

3. ○76%

2. [tex]\displaystyle See\:above\:grid[/tex]

1. [tex]\displaystyle See\:above\:grid[/tex]

Step-by-step explanation:

10. To convert from a percentage to a decimal, move the decimal mark twice to the left; each 20 is worth 5, and since 5 by 17 is 85, you have your fractional part of 17⁄20, then attach the whole number of 4.

9. To convert from a mixed number\improper fraction to a percentage, first evaluate the fractional part for a decimal answer, then move the decimal mark twice to the right.

8. To convert from a fraction to a percentage, evaluate the fraction for a decimal answer, then move the decimal mark twice to the right.

7. [tex]\displaystyle \frac{132}{220} = \frac{3}{5} =[/tex]60%

Greatest Common Divisor [GCD]: 44

6. [tex]\displaystyle \frac{11}{25} =[/tex]44%

_

[tex]\displaystyle \frac{4}{9} =[/tex]44,4%

[tex]\displaystyle 0,4 =[/tex]40%

Now that these are all percentages, it is alot easier to order them from least to greatest.

5. To convert from a percentage to a decimal, move the decimal mark twice to the left.

4. To convert from a decimal to a percentage, move the decimal mark twice to the right.

3. Each 25 is worth 4, and since 4 by 19 is 76, you get 76%.

2. Each 25 is worth 4, and since 4 by 6 is 24, you get 24%, and this graph.

1. Each 36 is worth 2 7⁄9, and since 12 by 2 7⁄9 is 33⅓, you get 33⅓%, so you would choose this answer.

I am joyous to assist you anytime.

Step-by-step explanation:

You must write formulas regarding the volume and surface area of ​​the given solids.

[tex]\bold{\#1\ Rectangular\ prism:}\\\\V=lwh\\SA=2lw+2lh+2wh=2(lw+lh+wh)\\\\\bold{\#2\ Cylinder:}\\\\V=\pi r^2h\\SA=2\pi r^2+2\pi rh=2\pir(r+h)\\\\\bold{\#3\ Sphere:}\\\\V=\dfrac{4}{3}\pi r^3\\SA=4\pi r^2[/tex]

[tex]\bold{\#4\ Cone:}\\\\V=\dfrac{1}{3}\pi r^2h\\\\\text{we need calculate the length of a slant length}\ l\\\text{use the Pythagorean theorem:}\\\\l^2=r^2+h^2\to l=\sqrt{r^2+h^2}\\\\SA=\pi r^2+\pi rl=\pi r^2+\pi r\sqrt{r^2+h^2}=\pi r(r+\sqrt{r^2+h^2})\\\\\bold{\#5\ Rectangular\ Pyramid:}\\\\V=\dfrac{1}{3}lwh\\\\[/tex]

[tex]\\\text{we need to calculate the height of two different side walls}\ h_1\ \text{and}\ h_2\\\text{use the Pythagorean theorem:}\\\\h_1^2=\left(\dfrac{l}{2}\right)^2+h^2\to h_1=\sqrt{\left(\dfrac{l}{2}\right)^2+h^2}=\sqrt{\dfrac{l^2}{4}+h^2}=\sqrt{\dfrac{l^2}{4}+\dfrac{4h^2}{4}}\\\\h_1=\sqrt{\dfrac{l^2+4h^2}{4}}=\dfrac{\sqrt{l^2+4h^2}}{\sqrt4}=\dfrac{\sqrt{l^2+4h^2}}{2}[/tex]

[tex]\\\\h_2^2=\left(\dfrac{w}{2}\right)^2+h^2\to h_2=\sqrt{\left(\dfrac{w}{2}\right)^2+h^2}=\sqrt{\dfrac{w^2}{4}+h^2}=\sqrt{\dfrac{w^2}{4}+\dfrac{4h^2}{4}}\\\\h_2=\sqrt{\dfrac{w^2+4h^2}{4}}=\dfrac{\sqrt{w^2+4h^2}}{\sqrt4}=\dfrac{\sqrt{w^2+4h^2}}{2}[/tex]

[tex]SA=lw+2\cdot\dfrac{lh_1}{2}+2\cdot\dfrac{wh_2}{2}\\\\SA=lw+2\!\!\!\!\diagup\cdot\dfrac{l\cdot\frac{\sqrt{l^2+4h^2}}{2}}{2\!\!\!\!\diagup}+2\!\!\!\!\diagup\cdot\dfrac{w\cdot\frac{\sqrt{w^2+4h^2}}{2}}{2\!\!\!\!\diagup}\\\\SA=lw+\dfrac{l\sqrt{l^2+4h^2}}{2}+\dfrac{w\sqrt{w^2+4h^2}}{2}\\\\SA=\dfrac{2lw}{2}+\dfrac{l\sqrt{l^2+4h^2}}{2}+\dfrac{w\sqrt{w^2+4h^2}}{2}\\\\SA=\dfrac{2lw+l\sqrt{l^2+4h^2}+w\sqrt{w^2+4h^2}}{2}[/tex]

Answer:

Y = 5x + 2 . . . . . . . . . . (1)

3x = -y + 10 . . . . . . . . (2)

3x = -(5x + 2) + 10

3x = -5x - 2 + 10

3x + 5x = -2 + 10

8x = 8

x = 1

y = 5(1) + 2 = 5 + 2 = 7

Solution is (1, 7)

Step-by-step explanation:

The Solution is (1,7) is the solution to the system of equations:

y=5x+2, 3x=-y + 10

Simplify simply means to make it simple. In mathematics, simply or simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.

here, we have,

given that,

We need to find the solution of the system of following equations.

y= 5x + 2        eq(1)

3x = -y +10      eq(2)

We will solve the equations using Substitution method to find the values of x and y

we put value of y from eq (1) into eq (2), The eq(2) will be:

3x = - (5x + 2) + 10

3x = -5x -2 +10

3x+5x = -2+10

8x = 8

x= 1

Now, putting value of z in eq(1) to find value of y

y = 5x +2

y = 5(1) + 2

y = 5+2

y = 7

So, Solution is (1,7).

Hence, The Solution is (1,7) is the solution to the system of equations:

y=5x+2, 3x=-y + 10

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Answer:

1.13 repeated

Step-by-step explanation:

Answer:

1 and 100 over 750

Step-by-step explanation:

Answer:

Volume of the cone in ascending order.

[tex]V_{2}=270\pi\ units^{3}<V_{3}=300\pi\ units^{3}<V_{4}=363\pi\ units^{3}<V_{1}=400\pi\ units^{3}[/tex]

cone with DIAMETER of 18 & height of 10

cone with RADIUS of 10 & height of 9

cone with RADIUS of 11 & height of 9

cone with DIAMETER of 20 & height of 12

Step-by-step explanation:

Let [tex]V_{2}. V_{3}. and\  V_{4}.[/tex] be the volume of the cone.

Let d, r and h be the diameter, radius and height of the cone.

Given:

[tex]d_{1} = 20\ and\ h_{1}=12[/tex]

[tex]d_{2} = 18\ and\ h_{2}=10[/tex]

[tex]r_{3} = 10\ and\ h_{3}=9[/tex]

[tex]r_{4} = 11\ and\ h_{14}=9[/tex]

Arrange the cones in order from lease volume to greatest volume.

Solution:

The volume of the cone is given below.

[tex]V=\pi r^{2} \frac{h}{3}[/tex]----------------(1)

where: r is radius of the base of cone.

and h is height of the cone.

The volume of the cone for [tex]d_{1} = 20\ and\ h_{1}=12[/tex]

[tex]r_{1} = \frac{d_{1}}{2}[/tex]

[tex]r_{1} = \frac{20}{2}=10\ units[/tex]

[tex]V_{1}=\pi (r_{1})^{2} \frac{h_{1}}{3}[/tex]

[tex]V_{1}=\pi (10)^{2} \frac{12}{3}[/tex]

[tex]V_{1}=\pi\times 100\times 4[/tex]

[tex]V_{1}=400\pi\ units^{3}[/tex]

Similarly, for volume of the cone for [tex]d_{2} = 18\ and\ h_{2}=10[/tex]

[tex]r_{2} = \frac{d_{2}}{2}[/tex]

[tex]r_{2} = \frac{18}{2}=9\ units[/tex]

[tex]V_{2}=\pi (r_{2})^{2} \frac{h_{2}}{3}[/tex]

[tex]V_{2}=\pi (9)^{2} \frac{10}{3}[/tex]

[tex]V_{2}=\pi\times 81\times \frac{10}{3}[/tex]

[tex]V_{2}=\pi\times 27\times 10[/tex]

[tex]V_{2}=270\pi\ units^{3}[/tex]

Similarly, for volume of the cone for [tex]r_{3} = 10\ and\ h_{3}=9[/tex]

[tex]V_{3}=\pi (r_{3})^{2} \frac{h_{3}}{3}[/tex]

[tex]V_{3}=\pi (10)^{2} \frac{9}{3}[/tex]

[tex]V_{3}=\pi\times 100\times 3[/tex]

[tex]V_{3}=\pi\times 300[/tex]

[tex]V_{3}=300\pi\ units^{3}[/tex]

Similarly, for volume of the cone for [tex]r_{4} = 11\ and\ h_{4}=9[/tex]

[tex]V_{4}=\pi (r_{4})^{2} \frac{h_{4}}{3}[/tex]

[tex]V_{4}=\pi (11)^{2} \frac{9}{3}[/tex]

[tex]V_{4}=\pi\times 121\times 3[/tex]

[tex]V_{4}=\pi\times 363[/tex]

[tex]V_{4}=363\pi\ units^{3}[/tex]

So, the volume of the cone in ascending order.

[tex]V_{2}=270\pi\ units^{3}<V_{3}=300\pi\ units^{3}<V_{4}=363\pi\ units^{3}<V_{1}=400\pi\ units^{3}[/tex]

Answer:

6*(3/4) is 75% of 6.

Step-by-step explanation:

3/4 =0.75

Consider this:

6*1 =6, so 1=100%.

6*1/2 =6*0.5=3, so 1/2=50%.

You can then see that 6*0.75=4.5 is 75% of 6.

Answer:

Step-by-step explanation:

Answer:

-32

Step-by-step explanation:

Divide both sides by 3/4 to solve for x.

－24/1*4/3 = -96/3

－96/3=－32

Answer:

Step-by-step explanation:

Use the formula

[tex]A(t)=P(1+\frac{r}{n})^{(n)(t)}[/tex]

where A(t) is the amount after all the compounding is done, P is the initial investment, r is the interest rate as a decimal, n is the number of times the investment is compounded each year, and t is time in years.  For us,

P = 6050

r = .046

n = 4

t = 6

A(t) = ?

Filling in our given info:

[tex]A(t)=6050(1+\frac{.046}{4})^{(4)(6)}[/tex]

which simplifies to

[tex]A(t)=6050(1+.0115)^{24}[/tex]

which simplifies a bit more to

[tex]A(t)=6050(1.0115)^{24}[/tex] and

A(t) = 6050(1.31577397) so

A(t) = $7960.43

which is choice C

Final answer:

The balance after 6 years is $6810.53 that is option A is correct.

Explanation:

To find the balance after 6 years, we can use the formula for compound interest: A = P(1 + r/n)^(nt).

Given that the initial principal (P) is $6050, the interest rate (r) is 4.6% (or 0.046 in decimal form), and it is compounded quarterly (n=4 times per year), we can plug in the values and solve for A.

A = $6050(1 + 0.046/4)^(4*6) = $6810.53

Therefore, the balance after 6 years is $6810.53, which corresponds to answer choice A.

Answer:

Point T

Step-by-step explanation:

The only point that is right across from (-3,-5) is point T. If you reflect across the Y-axis, you get (3,-5)

Answer:

324

Step-by-step explanation:

Given:

[tex]g(x)=4x^2+2x\\ \\f(x)=\int\limits^x_0 {g(t)} \, dt[/tex]

Find:

[tex]f(6)[/tex]

First, find f(x):

[tex]f(x)\\ \\=\int\limits^x_0 {g(t)} \, dt\\ \\=\int\limits^x_0 {(4t^2+2t)} \, dt\\ \\=\left(4\cdot \dfrac{t^3}{3}+2\cdot \dfrac{t^2}{2}\right)\big|\limits^x_0\\ \\=\left(\dfrac{4t^3}{3}+t^2\right)\big|\limits^x_0\\ \\= \left(\dfrac{4x^3}{3}+x^2\right)-\left(\dfrac{4\cdot 0^3}{3}+0^2\right)\\ \\=\dfrac{4x^3}{3}+x^2[/tex]

Now,

[tex]f(6)\\ \\=\dfrac{4\cdot 6^3}{3}+6^2\\ \\=288+36\\ \\=324[/tex]

Answer:

.45($259) = $116.55

The sale price of the surfboard is $116.55.

Answer:

The answer is k = [tex]-\frac{3}{5}[/tex]

Explanation:

We know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form  [tex]k=\frac{y}{x}[/tex] or [tex]y=kx[/tex]

In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin

we have the point (-30,18)

so

x=-30, y=18

Find the value of k

[tex]k=\frac{y}{x}[/tex]

substitute

[tex]k=\frac{18}{-30}[/tex]

Simplify

Divide by 6 both numerator and denominator

[tex]k=- \frac{3}{5}[/tex]

Answer:

Step-by-step explanation:

the amount of mark up = 23% of 80

= [tex]\frac{23*80}{100}[/tex]

= 23 * 0.8 = $18.4

Answer:

Only 1 time

Step-by-step explanation:

When it is compound interest it can be added in the following ways:

Annually = 1 Time in a year

Semiannually = 2 Times in a year

Quarterly = 4 Times in a year

Monthly = 12 Times in a year

Answer:

2

Step-by-step explanation:

a.p.e.x

The equation that relates y, the amount of yellow paint in liters, and b, the amount of blue paint in liters, needed to make the Green Goober's special green paint is y + b = 888

Solution:

Given that the Green Goober mixes 333 liters of yellow paint with 555 liters of blue paint to make 888 liters of special green paint for his costume.

Let "y" be the amount of yellow paint in liters needed to make the Green Goober's special green paint

Let "b" be the amount of blue paint in liters needed to make the Green Goober's special green paint

The required equation is:

amount of yellow paint in liters +  amount of blue paint in liters = Green Goober's special green paint

y + b = 888

Where y = 333 liters and b = 555 liters ( from given information)

Thus the equation is found

Answer:

y=3/5b

Step-by-step explanation:

Answer:

b. 2005

Step-by-step explanation:

(apex)

300=150(1.07)^x

1.07^x=2

x=ln 2/ln 1.07

=10.24

1995 plus ten years = 2005

hope this helps

Answer:

B. 2005

Step-by-step explanation:

We have been given that population of deer in a certain national park can be approximated by the function [tex]P(x)=150(1.07)^x[/tex], where x is the number of years since 1995. We are asked to find the year in which population will reach 300.

To solve our given problem, we will equate [tex]P(x)=300[/tex] and solve for x as:

[tex]300=150(1.07)^x[/tex]

[tex]\frac{300}{150}=\frac{150(1.07)^x}{150}[/tex]

[tex]2=(1.07)^x[/tex]

Now, we will take natural log on both sides as:

[tex]\text{ln}(2)=\text{ln}((1.07)^x)[/tex]

[tex]\text{ln}(2)=x\text{ln}(1.07)[/tex]

[tex]x=\frac{\text{ln}(2)}{\text{ln}(1.07)}[/tex]

[tex]x=10.2447[/tex]

[tex]x\approx 10[/tex]

Now, we will find 10 years after 1995 that is [tex]1995+10=2005[/tex].

Therefore, the population will be 300 in year 2005 and option B is the correct choice.

The solution to the given equation (-20+5) (58-65) is 105. This is calculated by first simplifying the expressions within the brackets and then multiplying the resulting numbers.

The student's question relates to the calculations and simplifying of expressions in mathematics. This type of operation can be found in basic algebra, and its mastery is an essential part of succeeding in mathematics.

To solve the equation given, which is (-20+5) (58-65), we'll need to separate it into two steps. First, simplify the expressions in the brackets. Therefore, -20+5 equals -15 and 58-65 equals -7. Now, substitute these values back into the equation getting -15 * -7. Multiplying these two values will give a product of 105. That is the final answer. The provided list of numbers following the initial problem statement seems to be irrelevant to this specific calculation.

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Solve for x in this equation.

x+(-9)=-36

x-9=-36

x=-27

answer: -27

Answer:

Circle the 15 in the diagram, that's the correct answer

Answer:

184π units²

Step-by-step explanation:

see attached for reference

given sloped height = 15 units

base radius = 8 units

the height of the cone can be found by using the Pythagorean equation

(sloped height)² = height ² + (base radius)²

15² = h² + 8²

h² = 15²-8² = 161

h = √161

Base surface area = πr² = π 8² = 64π units²

Lateral Surface Area,

= πr √(r² + h²)

= π(8) √[  (8)²  + (√161)²  ]

= 8π√(64 + 161)

= 8π√225

= 8π (15)

= 120π

Surface area = base area + lateral surface area

= 64π + 120π

= 184π units²

Answer:

The answer should be C. 1 1/4

Step-by-step explanation:

Eliminate A and B because none of them work

Eliminate E because we are using 1/4s

Eliminate D because we aren't using decimals

You can also think of a clock with adding 1/4s: 0.75 + 1/4= 1

1 + 1/4 = 1 1/4

(you can divide 1/2 into 2 one fourths)

The answer is 1 1/4 option C.

What is the sum of 1/2 and 0.75?

Simply add these two terms i.e:

1/2 + 0.75

Write 0.75 as a 75/100 and take lcm and lcm is 100.

1/2 + 75/100

On solving we get,

= ( 50 + 75 )/100

= 125/100

= 5/4

= 1 1/4

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The area common to both triangles is [tex]18\sqrt3[/tex] square centimeters.

In a 30-60-90 triangle, the ratio of the side lengths is [tex]1:\sqrt{3} :2[/tex]. Since the hypotenuse is 12 cm, we can determine the lengths of the other sides using this ratio.

The shorter leg (opposite the 30-degree angle) is (1/2) times the hypotenuse, which is (1/2) * 12 cm = 6 cm.

The longer leg (opposite the 60-degree angle) is [tex]\sqrt3[/tex] times the shorter leg, which is 6 * [tex]\sqrt{3[/tex] cm = [tex]6\sqrt3[/tex] cm.

Now, since the two triangles are congruent, the overlapping region forms an isosceles triangle with two sides measuring 6 cm (the shorter leg) and a base measuring [tex]6\sqrt3[/tex] cm (the longer leg).

The base of the isosceles triangle is [tex]6\sqrt3[/tex] cm, and since it's an isosceles triangle, the height is the same as the shorter leg, which is 6 cm.

Common Area = ([tex]6\sqrt3[/tex] cm * 6 cm) / 2 = [tex]36\sqrt3[/tex] cm² / 2 = [tex]18\sqrt3[/tex] cm².

Therefore, the area common to both triangles is [tex]18\sqrt3[/tex] square centimeters.

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The area common to both triangles is 144 square centimeters.

To find the area common to both triangles, we need to determine the overlapping region, which is a rhombus formed by the intersection of the two 30-60-90 triangles.

First, let's consider one of the 30-60-90 triangles. The sides of a 30-60-90 triangle are in the ratio 1:√3:2. In this case, the hypotenuse is 12 cm, so the sides of the triangle are:

Shorter leg (opposite the 30-degree angle) = 12 cm / 2 = 6 cm

Longer leg (opposite the 60-degree angle) = 6 cm * √3

Now, let's look at the overlapping region, which forms a rhombus. The diagonals of a rhombus are perpendicular bisectors of each other, so each diagonal will be twice the length of the shorter leg of the 30-60-90 triangle.

Diagonal of the rhombus = 2 * 6 cm = 12 cm

The area of a rhombus can be calculated using the formula: Area = (diagonal1 * diagonal2) / 2

In this case, both diagonals are equal (12 cm each), so the area of the rhombus (and hence the common area of the triangles) is:

Area = (12 cm * 12 cm) / 2

Area = 144 cm²

So, the area common to both triangles is 144 square centimeters.

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a. She pays $100 for 5 sheets

b. 25+15x dollars for x sheets

c. 8 sheets

Step-by-step explanation:

Given

Sitting cost = $25

Per sheet picture cost = $15

Let p be the number of sheets of pictures

Then the cost can be written as a function of p

[tex]c(p) = 25+15p[/tex]

Now,

a. How much does she pay for 5 sheets of pictures?!

Putting p = 5 in the function

[tex]c(5) = 25 + 15(5)\\= 25+75\\=100[/tex]

She will pay $100 for 5 sheets of pictures

b. How much does she pay for "x" sheets?

Putting x in place of p

[tex]c(x) = 25+15x[/tex]

c. How many sheets can she buy for $145?

We know the cost now, we have to find p so,

[tex]145 = 25+15p\\145-25 = 25+15p-25\\120 = 15p[/tex]

Dividing both sides by 15

[tex]\frac{15p}{15} = \frac{120}{15}\\p = 8[/tex]

Hence,

She can buy 8 sheets for $145

Keywords: Linear equation, Algebraic functions

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