MODELING A cyclist pedals at a rate of 6 rotations every 5 seconds. The motion of the pedals is circular with a radius of 7 inches. The closest the pedals get to the ground is four inches away. Write a function h(t) that models the height of a pedal in inches as a function of the time t in seconds. Assume the pedal starts at its highest point

A) h(t) = 5sin (12π\7 t)+11
B) h(t) = 12\7sin(5π t)+11
C) h(t) = 5cos(12π\7 t)+11
D) h(t) = 12\7 cos(5π t)+11

Using LCM it is fοund that bοth trams will be at the statiοn again in 72 minutes.

In mathematics, the least cοmmοn multiple is sοmetimes referred tο as LCM οr the lοwest cοmmοn multiple. The smallest number amοng all the cοmmοn multiples οf the prοvided numbers is the least cοmmοn multiple οf twο οr mοre numbers.

The trams will meet again at the statiοn when bοth their arrival times cοincide.

We can find this time by finding the least cοmmοn multiple (LCM) οf the twο arrival times.

The LCM οf 18 and 24 is 72.

Therefοre, bοth trams will meet again at the statiοn in 72 minutes.

Tο see why this is the case, cοnsider the fοllοwing arrival times fοr each tram -

Tram A: 0, 18, 36, 54, 72, 90, 108, ...

Tram B: 0, 24, 48, 72, 96, 120, ...

We can see that bοth trams arrive at the statiοn at the same time (72 minutes) in their 4th cycle οf arrivals (Tram A: 54, Tram B: 72).

Therefοre, the time value is οbtained as 72 minutes.

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Answer:

5/36 is the simplest answer

Step-by-step explanation:

i did the math and searched it up i hope this helps

The probabilities are given as follows:

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

For a probability tree, the sum of the nodes at each level must be of 1.

Then the value of a is obtained as follows:

0.6 + a = 1

a = 1 - 0.6

a = 0.4.

The value of b is obtained as follows:

0.9 + b = 1

b = 0.1.

The values of c and d are free, as long as the sum of these two values is of 1, hence we can attribute:

c = 0.5, d = 0.5.

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Answer:

The statement is correct

Step-by-step explanation:

Use the equation of a circle:

x² + y² = r²

x² + y² = 3²

then enter the coordinates of the point (2; √5):

2² + (√5)² = 9

4 + 5 = 9 (that is correct)

The point (2, √5) lies on the circle centered at the origin with radius 3 is True statement.

The standard equation of a circle is given by:

(x-h)² + (y-k)² = r²

Where (h,k) is the coordinates of center of the circle and r is the radius.

We have,

The point  (2, √5) lies on the circle centered at the origin with radius 3.

We know the equation of circle at origin is

x² + y² = r², where r is the radius of circle

So, x² + y² = 3²

Now. to check the point put x= 2 and y= √5 as

x² + y²

= 2² + √5²

= 4 +5

= 9

Thus, the given statement is Correct.

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Answer: To find the length of the other side of the rectangle, we can use the formula for the area of a rectangle, which is:

area = length x width

We know that the area is 2314 square miles and one side is 26 miles. Let's plug these values into the formula and solve for the other side:

2314 = 26 x width

Divide both sides by 26:

width = 2314/26

Simplify:

width = 89

Therefore, the other side of the rectangle is 89 miles.

Step-by-step explanation:

[tex]\begin{array}{llll} 19. & \text{x}= \frac{9\sqrt{6}}{2}, & \text{y}= \frac{9\sqrt{6}}{2}, & \text{z}= 18\\\\21. & \text{x}= 6, & \text{y}= 6\sqrt{3}, & \text{z}= 6\sqrt{6}\\\\23. & \text{x}= 10\sqrt{3}, & \text{y}= 5\sqrt{3}, & \text{z}= 15\\\\\end{array}[/tex]

===========================================================

Work Shown:

Problem 19

The triangle up top is a 30-60-90 triangle. The short leg is opposite the smallest angle 30 degrees. Double the short leg to get the hypotenuse.

z = 2*9 = 18.

The long leg is found like so

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{long leg} = 9\sqrt{3}\\\\[/tex]

These two formulas apply to 30-60-90 triangles only.

The long leg of the 30-60-90 triangle up top is the hypotenuse of the 45-45-90 triangle at the bottom.

For 45-45-90 triangles, we can say:

[tex]\text{hypotenuse} = (\text{leg})*\sqrt{2}\\\\\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}\\\\\text{leg} = \frac{\text{hypotenuse}*\sqrt{2}}{2}\\\\\text{leg} = \frac{9\sqrt{3}*\sqrt{2}}{2}\\\\\text{leg} = \frac{9\sqrt{3*2}}{2}\\\\\text{leg} = \frac{9\sqrt{6}}{2}\\\\[/tex]

This represents the lengths of x and y. For any 45-45-90 triangle, the two legs are the same length (the right triangle is isosceles).

--------------------------------

Problem 21

Focus on the 30-60-90 triangle up top.

The hypotenuse of 12 divides in half to get x = 12/2 = 6 as the short leg.

The long leg is [tex]6\sqrt{3}[/tex]

This is also the leg of the 45-45-90 triangle down below. Therefore, we have [tex]y = 6\sqrt{3}[/tex]

Multiply the leg by sqrt(2) to find the hypotenuse.

[tex]\text{hypotenuse} = \text{leg}*\sqrt{2}\\\\\text{z} = 6\sqrt{3}*\sqrt{2}\\\\\text{z} = 6\sqrt{3*2}\\\\\text{z} = 6\sqrt{6}\\\\[/tex]

--------------------------------

Problem 23

Focus on the triangle on the right. This is a 45-45-90 triangle.

The hypotenuse is 15sqrt(2), which must mean each leg is 15 units long. Therefore, z = 15. The unmarked vertical leg is also 15 units long.

This vertical side is the leg of the 30-60-90 triangle on the left. It's the long leg.

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{short leg} = \frac{\text{long leg}}{\sqrt{3}}\\\\\text{short leg} = \frac{(\text{long leg})*\sqrt{3}}{3}\\\\\text{y} = \frac{15\sqrt{3}}{3}\\\\\text{y} = 5\sqrt{3}\\\\[/tex]

Double this short leg to get the hypotenuse of the 30-60-90 triangle.

[tex]x = 2y\\\\x = 2*5\sqrt{3}\\\\x = 10\sqrt{3}\\\\[/tex]

Answer:

Question 19:

Question 21:

Question 23:

Step-by-step explanation:

A 45-45-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : 1 : √2.  Therefore, the formula for the ratio of the sides is b : b : b√2 where:

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.  Therefore, the formula for the ratio of the sides is c: c√3 : 2c where:

Side z is the hypotenuse of a 30-60-90 triangle where the leg opposite the 30° angle measures 9 units. Therefore, c = 9.

[tex]\implies z=2c =2 \cdot 9=18\; \sf units[/tex]

Therefore, the other leg of the same triangle (opposite the 60° angle) measures c√3 = 9√3 units.

Sides x and y are the congruent legs of a 45-45-90 triangle with hypotenuse measuring 9√3 units. Therefore b√2 = 9√3, so b = (9√6)/2.

[tex]\implies x=\dfrac{9 \sqrt{6}}{2}\; \sf units[/tex]

[tex]\implies y=\dfrac{9 \sqrt{6}}{2}\; \sf units[/tex]

Side x is the side opposite the 30° angle in a 30-60-90 triangle with a hypotenuse of 12 units. Therefore, 2c = 12 so c = 6.

[tex]\implies x=6\; \sf units[/tex]

Therefore, the other leg of the same triangle (opposite the 60° angle) measures c√3 = 6√3 units.

Side y is one of the congruent legs of a 45-45-90 triangle where the other congruent leg measures 6√3 units.

[tex]\implies y=6\sqrt{3}\; \sf units[/tex]

Side z is the hypotenuse of a 45-45-90 triangle where the congruent legs measure 6√3 units. Therefore b = 6√3.

[tex]\implies z=b\sqrt{2} = 6 \sqrt{3} \sqrt{2}=6\sqrt{6}\; \sf units[/tex]

Side z is one of the congruent legs of a 45-45-90 triangle where the hypotenuse measures 15√2 units. Therefore, b√2 = 15√2, so b = 15.

[tex]\implies z=b=15\; \sf units[/tex]

Side x is the hypotenuse of a 30-60-90 triangle where the leg opposite the 60° measures 15 units. Therefore, c√3 = 15, so c = 5√3.

[tex]\implies x=2c=2 \cdot 5\sqrt{3}=10\sqrt{3}\; \sf units[/tex]

Side y is the side opposite the 30° angle in a 30-60-90 triangle where the hypotenuse measures 10√3 units. Therefore, 2c = 10√3, so c = 5√3.

[tex]\implies y=c=5\sqrt{3}\; \sf units[/tex]

The cost per photo at which revenue is maximized is about $127.95.

We must first calculate the relevant y-value for the quadratic function at

x = 6.5 in order to get the pricing per photo that optimizes revenue (i.e., the revenue). When we enter x = 6.5, we obtain:

[tex]R(6.5) = 0.3(6.5)^2 + 3.9(6.5) + 97.5 =[/tex]$127.95

Before any costs or expenses are subtracted, revenue is the income derived from the sale of goods or services, or from any other use of capital or assets, related to an organization's primary operations. Also, it is known as sales or turnover.

from the question:

We can apply quadratic regression to identify the quadratic function that represents the revenue as a function of how many images are sold. This will provide us with the form's function:

[tex]R(x) = ax^2 + bx + c[/tex]

When x is the number of sold images, R(x), a, b, and c are coefficients we need to discover, and R(x) is the revenue.

We may construct an equation system to find the coefficients using the provided data:

(1) [tex]a(3)^2 + b(3) + c = 135[/tex]

(2)[tex]a(5)^2 + b(5) + c = 195[/tex]

(3) [tex]a(10)^2 + b(10) + c = 240[/tex]

When we simplify each equation, we obtain:

(1) 9a + 3b + c = 135

(2) 25a + 5b + c = 195

(3) 100a + 10b + c = 240

Any approach will work to solve this system of equations. We'll apply the substitution technique in this case.

Equation (1) gives us:

c = 135 - 9a - 3b

When we enter this into equation (2), we obtain:

25a + 5b + (135 - 9a - 3b) = 195

Simplifying, we get:

16a + 2b = 12

From equation (3), we get:

c = 240 - 100a - 10b

Substituting this into equation (1), we get:

9a + 3b + (240 - 100a - 10b) = 135

Simplifying, we get:

-91a - 7b = -105

Solving the system of equations 16a + 2b = 12 and -91a - 7b = -105, we get:

a = 0.3

b = 3.9

c = 97.5

Therefore, the quadratic function that models the revenue as a function of the number of photos sold is:

[tex]R(x) = 0.3x^2 + 3.9x + 97.5[/tex]

We must determine the x-value that corresponds to the parabola's vertex in order to determine the number of images that must be sold in order to maximize revenue. The vertex's x-value is determined by:

x = -b / 2a

Plugging in the values for a and b, we get:

x = -3.9 / 2(0.3) = -6.5

This indicates that the vertex of the parabola lies at x = 6.5 because the number of images sold cannot be negative. Darlena should therefore sell 6.5 photographs in order to increase her money.

We must first calculate the relevant y-value for the quadratic function at x = 6.5 in order to get the pricing per photo that optimizes revenue (i.e., the revenue). When we enter x = 6.5, we obtain:

[tex]R(6.5) = 0.3(6.5)^2 + 3.9(6.5) + 97.5 ≈ $127.95[/tex]

As a result, $127.95 per photo is the price that maximizes sales.

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Hence, x=-2 and y=4 are the answers to the system of equations.

For the purpose of connecting algebra and geometry with the aid of line and curve graphs, coordinate geometry is necessary. Finding points on a plane is a crucial component of mathematics. It also has a number of uses in other scientific fields such dimensional geometry, calculus, and trigonometry. Use y=-2x in place of x in the first equation:

5x - 9(-2x) = -46

Simplify and find x's value:

5x + 18x = -46

23x = -46

x = -2

We can use the second equation to get y now that we know the value of x:

y = -2x = -2(-2) = 4

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The name of the solid figure on the below object is indicated in front as follows:

Solid figures refers to three-dimensional objects with dimensions of length, width, and height. They have depth and take up space in our universe because they have three dimensions. The features that distinguish each type of solid are used to identify them.

In real life, solid figures include boxes, dice, tubes, traffic cones, balls, and tents. However, Ppisms, cubes, cones, spheres, pyramids, and cylinders are the most basic solid figures. A composite figure is formed by combining multiple solid figures.

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Answer:

Blank 1: incorrectly

Blank 2: no

Step-by-step explanation:

Blank 1:

The student did not properly distribute:

4( x + 4 ) = 4x + 16, NOT 4x + 4

4x + 4 = 4x + 16 [subtract 4x from each side]

4  = 16

An incorrect statement means that there are no solutions

The vertices of the image are: L' (1, -2), M' (2,-2), N' (2,2), P' (1, 2)

In mathematics and geometry, a vertex (plural: vertices) refers to a point where two or more lines, edges, or curves meet. In three-dimensional geometry, a vertex is typically the point where three or more edges or lines intersect to form a corner or a point.

[tex]r_{x-axis}[/tex] → Reflection across the x-axis

Transformation rule: (x, y) → (x,-y)

L(2, 4)→ [tex]L^{r}[/tex] (2,-4)

M (4,4) → [tex]M^{r}[/tex] (4-4)

N (4,-4) → [tex]N^{r}[/tex] (4, 4)

P (2-4) → [tex]P^{r}[/tex] (2, 4)

D0.5 →Dilation Centered at the origin by а scale factor of 0.5

Transformation rule: (x, y) → (0.5x,0.5y)

[tex]L^{r}[/tex] (2-4) → L' (1, -2)

[tex]M^{r}[/tex] (4,-4) → M' (2, -2)

[tex]N^{r}[/tex] (4, 4) → N' (2, 2)

[tex]P^{r}[/tex] (2, 4) → P' (1, 2)

The vertices of the image are:

L'(1, -2), M'(2,-2), N'(2,2), P' (1, 2)

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Answer:

f(x) = 2x^(2) - 3x + 6

f(0) = 2(0)^(2) - 3(0) + 6

f(0) = 0 - 0 + 6

f(0) = 6

Therefore, the value of f(0) is 6.

Step-by-step explanation:

Answer:

The equation of the given circle is:

(x + 3)² + (y + 7)² = 121

This equation is in standard form, which is:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is its radius.

Comparing the given equation with the standard form, we can see that:

The center of the circle is (-3, -7) (note the signs of h and k).

The radius of the circle is √121 = 11 (since r² = 121, then r = √121 = 11).

Therefore, the radius of the given circle is 11 units.

Step-by-step explanation:

Answer:

Afful is 21 and Naomi is 13.

Step-by-step explanation:

x=Afful's age

y=Naomi's age

System of equations:

x+5=2y

x+y=34

x+5=2y

y=(x+5)/2

x+y=34

x+((x+5)/2)=34

x=21

21+y=34

21+y=34

y=13

answer: Kevin received $30,000, and Bob received $70,000.

According to the problem, Bob received $10,000 more than two times what Kevin received. So, Bob received:

2X + $10,000

We also know that the total amount won is $100,000. Therefore, we can write the equation:

X + (2X + $10,000) = $100,000

Simplifying the equation, we get:

3X + $10,000 = $100,000

3X = $90,000

X = $30,000

Therefore, Kevin received $30,000, and Bob received:

2X + $10,000 = 2($30,000) + $10,000 = $70,000

So Bob received $70,000.

Answer: :)

An equation is something with a = sign and it’s C because it’s the only one with the sign

Step-by-step explanation:

The time until the minimum price is reached is 5 months. After this time, the price will start to increase again.

Price is the amount of money that must be paid to acquire a product or service, usually expressed in currency units such as dollars or euros.

The price of the computer system is given by the formula:

P = 170t² - 1700t + 6400

To find the time until the minimum price is reached, we need to find the value of t that corresponds to the vertex of the parabola described by this formula. The vertex is the point at which the price reaches its minimum value.

The formula for the x-coordinate of the vertex of a parabola in standard form (ax² + bx + c) is:

x = -b/2a

In this case, the formula for the price of the computer system is:

P = 170t² - 1700t + 6400

So we can see that a = 170, b = -1700, and c = 6400. Substituting these values into the formula for the x-coordinate of the vertex, we get:

t = -b/2a = -(-1700)/(2×170) = 5

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The next term of the sequence is -2

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

54, -18, 6, ...

The above definitions imply that we simply multiply -1/3 to the previous term to get the current term

Using the above as a guide,

so, we have the following representation

Next term = -1/3 * Current term

Substitute the known values in the above equation, so, we have the following representation

Next term = -1/3 * 6

Evaluate

Next term = -2

Hence, the next term is -2

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We can say that the arrοws pοint tο the variable term (3x) and the cοnstant term (-2) οf the expressiοn 3x - 2.

A linear expressiοn is a mathematical expressiοn in οne variable that can be written in the fοrm ax + b, where a and b are cοnstants and x is the variable. It represents a straight line when graphed.

The expressiοn 3x - 2 can be described as a sum οr difference οf twο terms: 3x and -2.

The cοefficient 3 is the numerical factοr οf the term 3x, which represents the prοduct οf 3 and x, a variable οr unknοwn quantity. The term 3x is called the variable term οr the first term οf the expressiοn.

The cοnstant term -2 is a term that dοes nοt invοlve any variables. It is the secοnd term οf the expressiοn and is alsο knοwn as the cοnstant term οr the cοefficient οf the cοnstant term, since it is multiplied by the cοefficient 1 (which is nοt written) in this case.

In οther wοrds, we can say that the arrοws pοint tο the variable term (3x) and the cοnstant term (-2) οf the expressiοn 3x - 2.

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the average nicotine content of one kind of cigarette exceeds that of another kind by 0.20 milligram at the 0.05 level of significance.

To test the hypothesis that the average nicotine content of one kind of cigarette exceeds that of another kind by 0.20 milligram, we need to compare the means of two independent samples. We can use a two-sample t-test for this purpose. The null hypothesis is that the difference between the means is equal to 0.20 milligram, and the alternative hypothesis is that it is not equal to 0.20 milligram.

The test statistic is given by:

t = ([tex]x_{1}[/tex] - [tex]x_{2}[/tex] - d) / [tex]\sqrt{s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2} }[/tex]

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and d is the hypothesized difference in means (0.20 milligram in this case).

Substituting the given values, we have:

x1 = 2.61 milligrams

[tex]x_{1}[/tex] = 2.38 milligrams

[tex]s_{1}[/tex] = 0.12 milligram

[tex]s_{2}[/tex] = 0.14 milligram

n1 = 50 cigarettes

n2 = 40 cigarettes

d = 0.20 milligram

t = (2.61 - 2.38 - 0.20) / [tex]\sqrt{0.12^{2}/50+0.14^{2}/40 }[/tex]= 4.57

The degrees of freedom for the test are given by:

[tex]df= (s_{1}^{2} /n_{1} +s_{2}^{2}/n_{2})^{2}/(s_{1}^{2} /s_{1})^{2}/_{1}-1 +(s_{2}^{2}/n_{2}^{2} /n_{2}-1[/tex])

Substituting the given values, we have:

df = [tex]( 0.12^{2}/50+0.14^{2}/40)^{2}/((0.12^{2}/50)^{2}/49+(0.14^{2}/40)^{2}/39= 86.36[/tex](rounded to two decimal places)

Using a t-distribution table or a calculator, we can find the critical values for a two-tailed test at the 0.05 level of significance with 86 degrees of freedom. For a two-tailed test, the critical values are ±1.987.

Since the calculated t-value of 4.57 is greater than the critical value of 1.987, we reject the null hypothesis.

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Answer:

0.3333333333 is right answer

Answer: Choice A  ​  

χ 2 =5.76;

0.10<P-value<0.15

Step-by-step explanation:

The equation of the line that is perpendicular to y = 2x + 4 and goes through the point (6, 4) is y = (-1/2)x + 7.

To find the equation of a line that is perpendicular to y = 2x + 4 and goes through the point (6, 4), we first need to determine the slope of the given line.

The equation y = 2x + 4 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Therefore, the slope of the given line is 2.

The slope of a line perpendicular to this one will be the negative reciprocal of 2, which is -1/2.

So the equation we are looking for will have a slope of -1/2 and will pass through the point (6, 4). We can use the point-slope form of the equation of a line to find it:

y - y1 = m(x - x1)

where m is the slope, (x1, y1) is the point, and x1 = 6, y1 = 4:

y - 4 = (-1/2)(x - 6)

Simplifying:

y - 4 = (-1/2)x + 3

y = (-1/2)x + 7

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Answer:

Step-by-step explanation:

Let the total number of beads in the box be x.

Then, the number of red beads in the box is 20% of x = 0.2x.

After increasing the number of red beads by 5%, the number of red beads becomes 1.05 times the original number of red beads, which is 1.05(0.2x) = 0.21x.

The total number of orange and white beads is (100% - 20%) = 80% of x, which is 0.8x.

After increasing the number of orange and white beads by half, the total number becomes 1.5 times the original number, which is 1.5(0.8x) = 1.2x.

So, we have the equation: 0.21x = 0.2x + 123 - 1.2x

Simplifying this equation, we get:

0.01x = 123

x = 12,300

Therefore, the box originally contained 12,300 beads.

The cost of having a car towed 14 miles is $103.

What is function ?

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range) with the property that each input is related to exactly one output. Functions are often represented by a formula or rule that describes how to compute the output value from the input value.

The towing cost, C, can be expressed as a function of miles towed, x, using the given information as follows:

C(x) = 75 + 2x

where x is the number of miles towed.

To find the cost of towing a car 14 miles, we can substitute x = 14 into the function:

C(14) = 75 + 2(14)

= 75 + 28

= $103

Therefore, the cost of having a car towed 14 miles is $103.

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Step-by-step explanation:

60-3x+8x=180

5x=180-60

5x/5=120/5

X=24

Answer:There’s no equation or numbers in your word problem

Step-by-step explanation:

The square yards the company needs to mow for the yard shape below is 58.88 square yards.

A square is a two-dimensional figure that has four sides and all four sides are equal.

The area of a square is given as side²

We have,

From the figure,

The yard shape has two shapes.

Triangle and a square.

Now,

Area of the triangle shape.

= 1/2 x 6.4 x 5.6

= 17.92 square yards

Area of the square shape.

= 6.4²

= 40.96 square yards

Now,

The square yards the company needs to mow for the yard shape below.

= 17.92 + 40.96

= 58.88 square yards.

Thus,

The square yards the company needs to mow for the yard shape below is 58.88 square yards.

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Answer:

≈ $5194,7

Step-by-step explanation:

First, find out how much did the price decrease:

[tex] \frac{7421 \times 30\%}{100\%} = 2226.3[/tex]

Now, subtract this number from the previous one (the price before):

$7421,00 - $2226,30 = $5194,70 ≈ $5194,7