What is the approximate area of the triangle?





A. 12. 5 square units


B. 18 square units


C. 21. 5 square units


D. 31 square units

Minimum

-11

Main concepts:

Concept 1: Identify the type of equation

Concept 2: Identify the concavity (opens up/down)

Concept 3: Finding a vertex of a parabola

Concept 1: Identify the type of equation

First, observe that the equation is a polynomial.  This is a type of equation where there may be multiple terms containing an x, where each term with an x is raised to a whole number power, and may be multiplied by a real number.  Additionally, there may be a constant term added (or subtracted).

For our equation, [tex]y=3x^2-12x+1[/tex], the first two terms contain an x, each raised to a whole number power, and are multiplied by a number.  Additionally, there is a constant added to the end of the equation.  Therefore, this is a polynomial.

The largest power of x in a polynomial is called the "degree" of the polynomial.  Since the largest power of x is 2, this is called a second degree polynomial.  Another common name for a second degree polynomial is a quadratic equation.

This quadratic equation is already in what is known as "Standard form" [tex]y=ax^2+bx+c[/tex]

Concept 2: Identify the concavity (opens up/down)

For quadratic equations, the graph of the equation will be a sort of "U" shape" called a parabola.  The parabola either opens up or down depending on the "leading coefficient" in the quadratic equation.

The "leading coefficient" of any polynomial is the constant number that is multiplied to x in the term with the highest power.  In this case, the leading coefficient is 3.

A parabola opens up or down in correspondence with the sign of the leading coefficient.  If the leading coefficient is positive, the parabola opens upward.  If the leading coefficient is negative, the parabola opens downward.

Since the leading coefficient is 3, the parabola for our example opens upward.  The branches of the "U" will go upward forever, without a maximum.  However, the bottom of the "U" will have a minimum value.  We are assigned to find this minimum value (how low it goes).

Concept 3: Finding a vertex of a parabola

To find the vertex of a parabola, with an equation in standard form, there are a few methods, but the most straightforward is to use the vertex formula:

[tex]h=\dfrac{-b}{2a}[/tex]

Where "h" is the x-coordinate of the vertex, and "a" and "b" are the coefficients from the quadratic equation: [tex]y=ax^2+bx+c[/tex]

[tex]h=\dfrac{-(-12)}{2(3)}[/tex]

[tex]h=\dfrac{12}{6}[/tex]

[tex]h=2[/tex]

So, the parabola will have a vertex with an x-coordinate of "2", meaning that the lowest point will be at a position that is 2 units to the right of the origin... however, we still don't know how high that minimum is.  Fortunately, the equation [tex]y=3x^2-12x+1[/tex] itself gives the relationship between any x-value and the y-value that is associated with it.

[tex]y=3x^2-12x+1[/tex]

[tex]y=3(2)^2-12(2)+1[/tex]

[tex]y=3*4+(-12)*2+1[/tex]

[tex]y=12+-24+1[/tex]

[tex]y=-11[/tex]

So, the vertex of the parabola is (2,-11).

The height of the vertex is -11, so the value of the minimum is -11.

Side note: "What is the value of the minimum" is a different question that "where is the minimum at".  The minimum is at 2.  The actual value of the minimum is -11.

The scope of both functions is all real numbers.

To compute the values of (f+g)(x) and (f-g)(x), we apply the addition and subtraction of two distinct functions, respectively:

(f+g)(x) = f(x) + g(x) = [tex](-2x + 4) + (3x^2) = 3x^2 - 2x + 4[/tex]

(f-g)(x) = f(x) - g(x) = [tex](-2x + 4) - (3x^2) = -3x^2 - 2x + 4[/tex]

The scope of both functions is all real numbers.

Subsequently, we evaluate the expressions for x = -3:

(f+g)(-3) = [tex]3(-3)^2 - 2(-3) + 4[/tex] = 3(9) + 6 + 4 = 27 +6 +4 = 37

(f-g)(-3) = [tex]-3(-3)^2 - 2(-3) + 4[/tex]= -3(9) + 6 + 4 = -27 + 6 + 4 = -17

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Gilberto brought $69.00 to the state fair.

Let's start by assigning variables to represent the unknown values in the problem:

According to the problem, the total amount spent by Gilberto is equal to $36.50, so we can set up an equation:

x + (1/2)x + (1/6)x = 36.5

Simplifying the equation, we can combine the like terms:

(5/6)x = 36.5

To solve for x, we can multiply both sides by the reciprocal of 5/6:

x = 36.5 / (5/6) = $43.80

So the cost of the pass is $43.80. Using the values we assigned earlier, we can find the cost of the souvenir and the burger:

Therefore, Gilberto spent $43.80 on the pass, $21.90 on the souvenir, and $7.30 on the burger, for a total of $43.80 + $21.90 + $7.30 = $73.00.

However, we are also told that Gilberto had $4.00 left over after buying these items.

So we can subtract that from the total amount spent to get the initial amount of money that Gilberto brought to the fair:

$73.00 - $4.00 = $69.00

Therefore, Gilberto brought $69.00 to the state fair.

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Answer: The movie was 2 hours and 20 minutes long.

Step-by-step explanation:

basic adding + subtracting

Answer:

Constant

Step-by-step explanation:

What is an exponential function?

An exponential function is a function with the general form y = abx, a ≠ 0, b is a positive real number and b ≠ 1. In an exponential function, the base b is a constant. The exponent x is the independent variable where the domain is the set of real numbers.

In this case, y=ab^x

where 0.983 is in our b term, which gives the meaning that number is our constant in this exponential function.

Using the formula for the area of a circle:

A = πr^2

A = π(4m)^2

A = 16π

A ≈ 50.3 m^2

Breaking the circle into equal sectors and rearranging the sectors as a parallelogram:

We can break the circle into 8 equal sectors, like this:

[IMAGE: circle with 8 equal sectors]

Each sector is 1/8th of the circle, so its angle is 45°. We can rearrange the sectors to form a parallelogram, like this:

[IMAGE: parallelogram made up of 8 sectors of the circle]

The base of the parallelogram is the same as the circumference of the circle, which is 2πr:

base = 2πr

base = 2π(4m)

base = 8π

The height of the parallelogram is the radius of the circle, which is 4m.

Now we can find the area of the parallelogram:

A = base × height

A = 8π × 4m

A = 32π

A ≈ 100.5 m^2

Finally, we can divide the area of the parallelogram by 8 to get the area of the circle:

A = (area of parallelogram) ÷ 8

A = (32π) ÷ 8

A = 4π

A ≈ 12.6 m^2

Therefore, the area of the circle is approximately 50.3 m^2 (using the formula) or 12.6 m^2 (using the parallelogram method).

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The test that should be run on the claim by Proctor & Gamble should be B. z-test of a proportion.

The alternative hypothesis indicates a c. left-tailed test.

The p - value is 0.2665. Unilever does not have sufficient evidence to reject Proctor & Gamble's claim, so A. no.

Conducting a z-test of a proportion is necessary in light of the proportion-based nature (at least half of the bars being 99.44% pure) of the inquiry, instead of means. Proctor & Gamble had asserted that over fifty percent of their bars exhibit 99.44% purity or better; Unilever wishes to investigate this assertion's accuracy.

Consequently, we run our test under null-hypothesis framework that considers fifty percent of bars possessing sufficient purity level and alternative hypothesis positing <50% do. The resultant course of action entails carrying out a left-tailed test.

The p - value. Find the test statistic:

z = ( 0. 4795 - 0.5) / √ ( ( 0.5 x (1 - 0.5) ) / 146)

z = - 0. 6236

z- table shows the p - value is 0. 2665 as a result.

With the p - value being higher than the normal significant level of 0. 05, Unilever should not reject Proctor & Gamble's claim.

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Answer:the radius of the hemisphere with a volume of 324 ft³ is approximately 6.3 feet (to the nearest tenth).

Step-by-step explanation:

The formula for the volume of a hemisphere is:

V = (2/3) × π × r³, where V is the volume and r is the radius of the hemisphere.

We have been given the volume of the hemisphere as 324 ft³, so we can substitute this into the formula:324 = (2/3) × π × r³

To find the radius r, we need to solve for it. Dividing both sides by (2/3) × π gives:r³ = (324 / ((2/3) × π))r³ = (324 × 3) / (2 × π)r³ = 486 / π

Taking the cube root of both sides gives:r = (486 / π)^(1/3)

Using a calculator to evaluate this expression, we get:r ≈ 6.3

If after 2.00 minutes, Ethan is 195km away from his house towards west, Ethan's velocity is approximately 6393 km/h.

To find Ethan's velocity, we need to first determine the distance he traveled and the time he spent traveling.

Given:
1. Initial position: Ethan's house
2. Distance to park: 85 km south
3. Final position: 195 km west from house after 2 minutes

To find the total distance, we can use the Pythagorean theorem, as the path forms a right triangle:
Distance = √(85² + 195²) = √(7225 + 38025) = √(45250) ≈ 212.72 km

Now, let's convert the time from minutes to hours:
2 minutes = 2/60 hours ≈ 0.0333 hours

Finally, we can calculate Ethan's velocity:
Velocity = Distance / Time = 212.72 km / 0.0333 hours ≈ 6393 km/h

Ethan's velocity is approximately 6393 km/h.

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The mean value Taylor would use is 10.23 seconds.

To calculate the mean value for Taylor's recorded times, we add up the individual times (10 seconds, 10.3 seconds, and 10.4 seconds) to obtain a total of 30.7 seconds.

we divide this total by the number of trials, which in this case is 3.

30.7 seconds divided by 3 equals approximately 10.23 seconds.

The mean value of Taylor's recorded times for her car's trials is approximately 10.23 seconds.

The mean value is often used as a measure of central tendency to represent the average of a set of values.

In this case, it represents the average time recorded by Taylor during her trials.

By calculating the mean, we can compare this value with the mean times of other cars to assess performance.

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The domain of the function is 0 ≤ b ≤ 6, and the range of the function is 30 ≤ c ≤ 174.

The function that models the total weight of a crate with b boxes inside is given as:

c = 24b + 30

We know that each crate can hold a maximum of 6 boxes. Therefore, the number of boxes inside the crate can only take values from 0 to 6.

Domain:

The number of boxes b can take values from 0 to 6. Therefore, the domain of the function is:

0 ≤ b ≤ 6

Range:

To find the range of the function, we need to consider the maximum and minimum values that c can take when

0 ≤ b ≤ 6.

When b = 0, the crate is empty, and the total weight of the crate is:

c = 24(0) + 30 = 30.

When b = 6, the crate is full with 6 boxes, and the total weight of the crate is:

c = 24(6) + 30 = 174.

Therefore, the range of the function is:

30 ≤ c ≤ 174

We can then say the domain of the function is 0 ≤ b ≤ 6, and the range of the function is 30 ≤ c ≤ 174.

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Since the function approaches the same value (0) along both paths, we can claim that the limit lim(x,y)→(0,0) f(x,y) exists and is equal to 0.

Your question is asking whether the limit of the function f(x,y) exists at the point (0,0). The function f(x,y) is defined as:
f(x,y) = x^2 + y if (x,y) ≠ (0,0)
f(x,y) = 0 if x = y = 0
To verify whether the limit exists, we need to check if the function approaches a unique value as (x,y) approaches (0,0). In other words, we need to determine if lim(x,y)→(0,0) f(x,y) exists.
To verify this claim, consider the function along different paths towards (0,0). Let's examine two paths:
1) x = 0: As x approaches 0, f(0,y) = y, and the limit becomes lim(y→0) y = 0.
2) y = x: As y approaches 0 along this path, f(x,x) = x^2 + x, and the limit becomes lim(x→0) (x^2 + x) = 0.

Since the function approaches the same value (0) along both paths, we can claim that the limit lim(x,y)→(0,0) f(x,y) exists and is equal to 0.

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Answer:

1243 ft³

Step-by-step explanation:

Given the volume formula for a cylinder:

[tex]V=\pi r^{2} h[/tex]

and we know that the radius is 6 and the height is 11, we can substitute:

[tex]V=\pi 6^{2} 11[/tex]

square 6

V=π36(11)

use 3.14 for pi and multiply everything together

V=3.14(36)(11)

simplify

V=1243.44

1243.44 rounded to the nearest whole number is 1,243 ft³.

Hope this helps! :)

We can use the ratio test to determine whether the series ∑(n=1[tex])^(infinity)[/tex][tex]13^n[/tex]/ [(n+1)[tex]6^(2n+1)[/tex]] is convergent or divergent.

We evaluate the limit:

lim n → ∞ [tex]|(13^(n+1) / [(n+2)6^(2n+3)]) / (13^n / [(n+1)6^(2n+1)])|[/tex]

= lim n → ∞ [tex]|(13^(n+1) / 13^n) * [(n+1)6^(2n+1) / (n+2)6^(2n+3)]|[/tex]

= lim n → ∞ [tex]|13 / 6^2| * |(n+1)/(n+2)|[/tex]

= 13/36

Since the limit is less than 1, by the ratio test, the series is convergent.

To find the value of a_n, we can substitute n = 1 into the formula:

a_n = [tex]13^n / [(n+1)6^(2n+1)][/tex]

a_1 = [tex]13 / [(1+1)6^(2+1)] = 13 / (2*6^3)[/tex]

Therefore, a_1 = 13 / 432.

Note that we only needed to find the value of a_1 to apply the ratio test and determine the convergence of the series.

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Answer:

2.2

Step-by-step explanation:

To find your GPA, use the attached image:

After you've written down your numerical scores, divide it by the total classes you're taking, which is 7.

So, your GPA is 2.2

The volume of the solid obtained by rotating the region about the line x=9 is approximately 201.06 cubic units.

To find the volume of the solid obtained by rotating the region bounded by 2 =8 32? and 2 = -2y about the line x= 9, we can use the cylindrical shell method.

First, we need to sketch the region and the line of rotation:

  |         +---------+

8 |         |         |

  |         |         |

  |         +---------+ x=9

  |              

0 +---------------+

  0      4      8

The region is a rectangle with height 4 and width 8, centered at the origin. The line of rotation is x=9.

Now, we can express the volume of the solid as a sum of cylindrical shells:

V = ∫[0,4] 2πr h dx

where r is the distance between x=9 and the boundary of the region at height x, and h is the thickness of the shell.

Since the region is symmetric about the y-axis, we can consider only the right half of the region and multiply the result by 2 to get the total volume.

The equation of the boundary at height x is:

2 = -2y

y = -x/2

The distance between x=9 and this line is:

r = 9 - (-x/2) = 9 + x/2

The thickness of the shell is dx.

Substituting these values into the integral, we get:

V = 2 ∫[0,4] 2π(9 + x/2) dx

V = 2π ∫[0,4] (18 + x) dx

V = 2π [18x + (1/2)[tex]x^2[/tex]] from x=0 to x=4

V = 2π [(18*4 + (1/2)[tex]4^2[/tex]) - (180 + (1/2)*[tex]0^2[/tex])]

V = 64π ≈ 201.06

Therefore, the volume of the solid obtained by rotating the region about the line x=9 is approximately 201.06 cubic units.

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X represents the number of basketballs in Chris's gym class.

Chris represents the number of students in his gym class with the expression 4x + 2. We know that the number of students is four times the number of basketballs, so we can set up the equation 4x = the number of basketballs.

If we substitute this expression for the number of students in the gym class, we get 4(4x) + 2 = the total number of students in the gym class and locker room. We also know that there are two more students in the locker room, so we can add 2 to this expression to get 4(4x) + 4 = the total number of students in the gym class and locker room.

Now we can set this expression equal to the original expression for the number of students, 4x + 2, and solve for x:

4(4x) + 4 = 4x + 2

16x + 4 = 4x + 2

12x = -2

x = -1/6

However, x cannot represent a negative number of basketballs, so there must be an error in the problem.

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Solving for dz/dt, we get:

dz/dt ≈ 11.650 ft/sec.

So, after another minute, the distance between Claire and Anna is changing at a rate of approximately 11.650 ft/sec.

Hi there! To answer this question, we can use the Pythagorean theorem and implicit differentiation. Let x be the distance Claire runs east and y be the distance Anna runs north. After 1 minute, Claire has already run 12 * 60 = 720 ft. After another minute, x = 720 + 12t, and y = 7t.

Now, we can set up the Pythagorean theorem: x^2 + y^2 = z^2, where z is the distance between them. Substituting the expressions for x and y, we get (720 + 12t)^2 + (7t)^2 = z^2.

To find the rate at which the distance between them is changing (dz/dt), we need to differentiate both sides of the equation with respect to time, t:

2(720 + 12t)(12) + 2(7t)(7) = 2z(dz/dt).

Now, we can plug in the values for t = 2 minutes:

2(720 + 24)(12) + 2(14)(7) = 2z(dz/dt).

Simplifying, we get:

34560 + 392 = 2z(dz/dt).

After 2 minutes, Claire has run 12(120) = 1440 ft, and Anna has run 7(60) = 420 ft. Using the Pythagorean theorem, we can find z:

z = √(1440^2 + 420^2) ≈ 1500 ft.

Now we can find dz/dt:

34952 = 2(1500)(dz/dt).

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The linear equation in slope intercept form is:

y = -0.1x + 9

The formula for finding the Linear Equation in slope intercept form is expressed in the form:

y = mx + c

where:

m refers to the slope

c refers to the y-intercept

Looking at the given graph, we can see that:

The y-intercept = 9

The y-intercept is the point where the line crosses the y-axis while x-intercept is the point where the line crosses the x-axis.

Taking the two coordinates:

(1970, 7) and (1990, 5)

Slope:

m = (5 - 7)/(1990 - 1970)

m = -2/20

m = -0.1

Thus, the Equation in slope intercept form is expressed in the form of:

y = -0.1x + 9

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The height of a 16-year-old boy in the 96th percentile is approximately 73.375 inches. The z-score associated with the 96th percentile is 1.75.

To find the z-score associated with the 96th percentile, we can use the following steps:

1. Locate the percentile in a standard normal distribution table or use a calculator that can compute percentiles (e.g., a graphing calculator or an online calculator).

2. For the 96th percentile, we find the corresponding z-score. Using a standard normal distribution table or an online calculator, the z-score is approximately 1.75.

So, the z-score associated with the 96th percentile is 1.75.

To find the height of a 16-year-old boy in the 96th percentile, we can use the following formula:

Height = mean + (z-score × standard deviation)

Here, the mean height is 68.3 inches, the z-score is 1.75, and the standard deviation is 2.9 inches. Plugging these values into the formula:

Height = 68.3 + (1.75 × 2.9) ≈ 68.3 + 5.075 ≈ 73.375 inches

So, the height of a 16-year-old boy in the 96th percentile is approximately 73.375 inches.

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The measure of the largest interior angle is  105°.

What is the measure of angle?

When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.

Here, we have

Given: The measures of 5 of the interior angles of a hexagon are: 130, 120°, 80, 160, and 155. we have to find the measure of the largest interior angle.

The sum of all the six interior angles of a hexagon is 720°.

As sum of five angles is 130° + 120° + 80° + 160° +155° = 165°

The sixth angle is 720° - 165° = 75°

So the smallest interior angle of the hexagon is 75°.

and the largest exterior angle is 180° -  75° =  105°.

Hence, the measure of the largest interior angle is  105°.

Question: The measures of 5 of the interior angles of a hexagon are: 130, 120°, 80, 160, and 155 What is the measure of the largest exterior angle?

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The length of AB is approximately 12.704 units.

To solve this problem, we can use trigonometry and the fact that the easel forms a 30° angle to find the length of AB.

We know that RC is 22, and that angle R is 30°. Let's use the trigonometric function tangent to find AB:

tan(30°) = AB / RC

We can rearrange this equation to solve for AB:

AB = tan(30°) * RC

Using a calculator or trigonometric table, we find that tan(30°) = 0.5774 (rounded to four decimal places). Therefore:

AB = 0.5774 * 22

AB ≈ 12.704

So the length of AB is approximately 12.704 units.

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The function is increasing for all real values of x where x < –4.

The statement that is true about the function is: "The function is decreasing for all real values of x where x < -4."

In order to determine the behavior of the function, we look at the given options. Among the options, the only statement that aligns with the function being decreasing is the one that states the function is decreasing for all real values of x where x < -4.

If a function is decreasing, it means that as the value of x decreases, the value of the function also decreases. In this case, it indicates that as x becomes more negative, the function's values decrease.

Therefore, the statement that correctly describes the behavior of the function is that it is decreasing for all real values of x where x < -4.

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The option that can be true for the function g(x) is C; g(-13) = 20

Here we know that the function g(x) has:

The domain ---> -20 ≤ x ≤ 5

The range --->  -5 ≤ g(x) ≤ 45

And g(0) = -2

       g(-9) = 6

There are two statements that could be true:

g(-13) = 20, because -13 belongs to the domain and 20 belongs to the range.

g(0) = 2  could also be true.

Now, we can see that g(-9) > g(0), then as x becomes smaller, g increases, then the option that seems to be correct is g(-13) = 20

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Answer:

1. [tex]9^{2} + 12^2 = 15^2\\81+144=225[/tex]

32/9

So the producer surplus at the equilibrium point is 32/9 dollars.

To find the equilibrium point, we need to set D(x) equal to S(x) and solve for x:

(x-7)^2 = x^2 + 2x + 33

Expanding and simplifying:

x^2 - 14x + 49 = x^2 + 2x + 33

12x = 16

x = 4/3

So the equilibrium point is x = 4/3.

To find the consumer surplus at the equilibrium point, we need to find the difference between the maximum price consumers are willing to pay (D(4/3)) and the equilibrium price (S(4/3)) and multiply by the quantity sold (4/3):

Consumer surplus = (D(4/3) - S(4/3)) * (4/3)

= [(4/3 - 7)^2 - (4/3)^2 - 2(4/3) - 33] * (4/3)

= [49/9 - 16/9 - 8/3 - 33] * (4/3)

= -224/27

So the consumer surplus at the equilibrium point is -224/27 dollars.

To find the producer surplus at the equilibrium point, we need to find the difference between the equilibrium price (S(4/3)) and the minimum price producers are willing to accept (S(0)) and multiply by the quantity sold (4/3):

Producer surplus = (S(4/3) - S(0)) * (4/3)

= [(4/3)^2 + 2(4/3) + 33 - 33] * (4/3)

= 32/9

So the producer surplus at the equilibrium point is 32/9 dollars.

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Answer:

The answer to your problem is, 39

Step-by-step explanation:

In order to find the area of the triangle use the formula down below:

A = [tex]\frac{h_{b} b}{2}[/tex]

Base = 13

Height = 6

Replace them equals:

= [tex]\frac{6*13}{2}[/tex] = 39

Thus the answer to your problem is, 39

The simplified expression of 100-3(4. 25)-13-4(2. 99) is 48.29.

What is PEMDAS?

PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It is a mnemonic or acronym used to remember the order of operations when simplifying mathematical expressions.

To simplify the expression 100-3(4.25)-13-4(2.99), you can follow the order of operations (PEMDAS) which is:

Parentheses

Exponents

Multiplication and Division (from left to right)

Addition and Subtraction (from left to right)

Using this order, you can simplify the expression as follows:

100 - 3(4.25) - 13 - 4(2.99)

= 100 - 12.75 - 13 - 11.96 // multiply 3 and 4 with their respective numbers

= 62.29 - 13 - 11.96 // perform subtraction within parentheses

= 48.29 // perform final subtraction

Therefore, the simplified expression is 48.29.

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The amount of water in the tank can be calculated by multiplying 3/4 to the volume of the tank: 3/4 x V = (3/4)L x W x H.

To calculate the amount of water in the aquarium's fish tank in the shape of a prism,

you would need to know the dimensions of the tank and then multiply the volume of the tank by 3/4.

Let's assume that the aquarium has a rectangular prism shape,

the amount of water in the tank would depend on the dimensions of the tank.

Let's assume the tank has a length of L, a width of W, and a height of H.

The volume of the tank can be calculated by multiplying the length, width, and height together: V = L x W x H.

If the tank is 3/4 full of water, the volume of water in the tank would be 3/4 of the total volume of the tank.

Learn more about  rectangular prism here:

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