solve each system by substitution

y=-2x-7
2x-8y=-16

If in Angle STU, the measure of U=90°, the measure of S=31°, and TU = 77 feet, then the length of US to the nearest tenth of a foot is approximately 39.4 feet.

In angle STU, we have a right triangle with U=90°, S=31°, and TU=77 feet. To find the length of US, we can use the sine function:

sin(S) = opposite side (US) / hypotenuse (TU)

sin(31°) = US / 77 feet

To find the length of US, multiply both sides by 77 feet:

US = 77 feet * sin(31°)

US ≈ 39.4 feet

Therefore, the length of US to the nearest tenth of a foot is approximately 39.4 feet.

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The volume of cement in cubic feet that will be needed is 45 cubic feet.

Volume is the space occuppied by an object.

To calculate the volume of cement in cubic feet that will be needed, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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Comets could be visible from Earth when they are most likely to fall down into earth

Answer:

Step-by-step explanation:

If there are 180 degrees in a triangle total and in this problem we know that one angle is 49 and the other is 39, we can assume that subtracting 39 and 49 from 180 will find x. In this case, x will be 92.

Answer:

a and b

Step-by-step explanation:

The volume of the cylinder is 4004 cubic metres.

The diameter of the cylinder is 28 metres and the height of the cylinder is 6.5 metres.

Therefore, the volume of the cylinder can be found as follows:

Hence,

volume of a cylinder = πr²h

where

Therefore,

volume of the cylinder =  22 / 7 × 14² × 6.5

volume of the cylinder = 22 / 7 × 196 × 6.5

volume of the cylinder = 28028 / 7

volume of the cylinder = 4004 cubic metres

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The equation of the hyperbola is -(x + 5)²/71 + (y - 2)² = -71

We can also find the distance between the center and the given focus, which is the distance between (-5, 2) and (-5 - √29, 2):

d = |-5 - (-5 - √29)| = √29

Substituting in the known values, we get:

c² = a² + b²

(√29)² = (10)² + b²

29 = 100 + b²

b² = -71

(x - h)²/a² - (y - k)²/b² = 1

where (h, k) is the center of the hyperbola.

Substituting in the known values, we get:

(x + 5)²/100 - (y - 2)²/-71 = 1

Multiplying both sides by -71, we get:

-(x + 5)²/71 + (y - 2)²/1 = -71/1

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Step-by-step explanation:

g(x)  is just f(x) shifted UP three units ...so

  g(x) = f(x) +3

The point (–2, 5) is located above the point (–2, –1).

The point (1, 212) is located to the left of the point (4, 212).

The point (3, –6) is located below the point (3, –3).

The point (−212, 1) is located to the left of the point (–3, 1).

The point (312, 12) is located to the right of the point (12, 12).

The point (2, 5) is located above the point (2, –5).

a. (–2, 5) is above (–2, –1). The two points have the same x-coordinate, but different y-coordinates. Since the y-coordinate increases as you move up on the coordinate plane, the point (–2, 5) is located above the point (–2, –1).

b. (1, 212) is to the left of (4, 212). The two points have the same y-coordinate, but different x-coordinates. Since the x-coordinate increases as you move to the right on the coordinate plane, the point (1, 212) is located to the left of the point (4, 212).

c. (3, –6) is below (3, –3). The two points have the same x-coordinate, but different y-coordinates. Since the y-coordinate decreases as you move down on the coordinate plane, the point (3, –6) is located below the point (3, –3).

d. (−212, 1) is to the left of (–3, 1). The two points have the same y-coordinate, but different x-coordinates. Since the x-coordinate decreases as you move to the left on the coordinate plane, the point (−212, 1) is located to the left of the point (–3, 1).

e. (312, 12) is to the right of (12, 12). The two points have the same y-coordinate, but different x-coordinates. Since the x-coordinate increases as you move to the right on the coordinate plane, the point (312, 12) is located to the right of the point (12, 12).

f. (2, 5) is above (2, –5). The two points have the same x-coordinate, but different y-coordinates. Since the y-coordinate increases as you move up on the coordinate plane, the point (2, 5) is located above the point (2, –5).

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To find the length of AC in triangle ABC, we will use the Angle Bisector Theorem and the given information:
In triangle ABC, the bisector of angle A divides BC into segments BD and DC, with lengths of 28 units and 24 units, respectively. Given that AB has a length of 31.5 units, we want to determine the possible length of AC.

Step 1: Apply the Angle Bisector Theorem, which states that the ratio of the lengths of the sides is equal to the ratio of the lengths of the segments created by the angle bisector. In this case, we have:

AB / AC = BD / DC

Step 2: Plug in the known values:

31.5 / AC = 28 / 24

Step 3: Simplify the ratio on the right side:

31.5 / AC = 7 / 6

Step 4: Cross-multiply to solve for AC:

6 * 31.5 = 7 * AC

Step 5: Calculate the result:

189 = 7 * AC

Step 6: Divide both sides by 7 to find AC:

AC = 189 / 7

Step 7: Calculate the value of AC:

AC = 27 units

So, the length of AC in triangle ABC could be 27 units.

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The rotation used in this problem is given as follows:

90º clockwise rotation.

The five more known rotation rules are given as follows:

A vertex and it's equivalent is given as follows:

A(1,3) and A'(3, -1).

Hence the rule is:

(x,y) -> (y, -x).

Which is the rule for a 90° clockwise rotation = 270º counterclockwise rotation.

The image is presented at the end of the answer.

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Answer:

90° clockwise rotation

Step-by-step explanation:

I did the exam and got it correct

Answer:

$264,000

Step-by-step explanation:

PV = $110,000

i = 6%

n = 15 years

Compound formula:

FV = PV (1 + i)^n

FV = 110,000 (1 + 0.06)^15

FV = 110,000 · 2.40(rounded) = $264,000

The student used 30.0 mL of 3.00 M HCl to make the 50.0 mL sample of 1.80 M HCl.

To find the volume of 3.00 M HCl needed to make a 50.0 mL sample of 1.80 M HCl, we can use the equation:

M₁V₁ = M₂V₂

Where M₁ is the initial concentration, V₁ is the initial volume, M₂ is the final concentration, and V₂ is the final volume.

We are given M₁ = 3.00 M, M₂ = 1.80 M, and V₂ = 50.0 mL. We can rearrange the equation to solve for V₁:

V₁ = (M₂V₂) / M₁

V₁ = (1.80 M * 50.0 mL) / 3.00 M

V₁ = 30.0 mL

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Approximate amount of dough that is remaining. Is  (length - 2r)(width - 3r) - 6πr^2.

Without the size of the original sheet of dough or the size of the circles cut from it, it's not possible to give an exact expression. However, assuming that each circle has the same radius of 'r' and the original sheet of dough was a rectangle, we can write an expression in factored form for the remaining area of the dough:

Remaining area of dough = (Area of original rectangle) - 6(Area of circle)

= (length x width) - 6(πr^2)

= (length - 2r)(width - 3r) - 6πr^2

Whether there is enough dough for another circle would depend on the size of the circles and the original sheet of dough.

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To minimize your losses while gambling at a casino and playing slot machines, you need to minimize your regret R in the experts problem. R is defined as the difference between your total cost and the best expert's cost.

To minimize R, follow these steps:

1. Begin by assigning equal weight to each expert (slot machine).
2. After each day t, observe the cost c_ti for each expert i.
3. Update the weights by multiplying them by (1 - c_ti), making sure they remain non-negative.
4. Normalize the weights so they sum up to 1.
5. On day t+1, choose the expert with the highest weight to take advice from.

By following this adaptive strategy, you will minimize your regret R, allowing you to reduce your losses while gambling at the slot machines.

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Answer:

Step-by-step explanation:

To find the solution set to the inequality w - 10 < 16, we can solve for w by adding 10 to both sides of the inequality:

w - 10 + 10 < 16 + 10 w < 26

This means that any number less than 26 is part of the solution set to the inequality. So, out of the given options, the number that is not part of the solution set is D. 27 because it is greater than 26.

Answer:

Yes

Step-by-step explanation:

Yes, because its graph represents a straight line

Answer:

A′(−2.5, 3.75), B′(−1.25, 1.25), C′(2.5, −1.25), D′(2.5, 2.5)

Step-by-step explanation:

in the described situation you only need to multiply the coordinates by the scale factor (in our case the given 5/8)

A (-4, 6) turns into

A' (-4×5/8, 6×5/8) = A' (-2.4, 3.75)

and therefore we know already here that all the other answer options are wrong.

The decay of the radioactive substance can be modeled by the exponential decay function:

y = a(1 - r)^t

where:

- y is the amount of substance present after t hours

- a is the initial amount of substance (in grams), which is 120 grams in this case

- r is the decay rate per hour, which is 18% or 0.18 in decimal form

- t is the time elapsed in hours

Plugging in the values we get:

y = 120(1 - 0.18)^t

Simplifying:

y = 120(0.82)^t

So this is the equation that represents the number of grams, y, present after t hours, given the initial amount of 120 grams and a decay rate of 18% per hour.

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There are infinitely many possible solutions for the lengths of the three sides of the triangle.

What is triangle?

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry.

If we call x the length of the smallest side, then the other two sides are x+1 and x+2 (since they are three consecutive integers). According to the triangle inequality, the sum of any pair of sides must be greater than the length of the third side.

Therefore, we have:

x + (x+1) > (x+2) (and also x + (x+2) > (x+1) and (x+1) + (x+2) > x)

Simplifying each inequality, we get:

2x + 1 > x + 2 (and also 2x + 2 > x + 1 and 2x + 3 > x)

Which gives:

x > 1

So the smallest side must be greater than 1 cm.

Now, to find the length of the three sides, we can choose any value greater than 1 for x. For example, if we take x=2, then the three sides are:

2 cm, 3 cm, and 4 cm

If we take x=3, then the three sides are:

3 cm, 4 cm, and 5 cm

And so on. Therefore, there are infinitely many possible solutions for the lengths of the three sides of the triangle.

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We can say with 95% confidence that the true mean cholesterol content of all such eggs is between 223.99 milligrams and 232.01 milligrams.

To construct a 95% confidence interval for the true mean cholesterol content of all such eggs, we can use the following formula:

CI = X ± Zα/2 * σ/√n

where:

X = sample mean = 228 milligrams

Zα/2 = the critical value from the standard normal distribution corresponding to a 95% confidence level, which is 1.96

σ = population standard deviation = 19.0 milligrams

n = sample size = 82

Substituting the values into the formula, we get:

CI = 228 ± 1.96 * 19.0/√82

= 228 ± 4.01

= (223.99, 232.01)

Therefore, we can say with 95% confidence that the true mean cholesterol content of all such eggs is between 223.99 milligrams and 232.01 milligrams.

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S = lim[n → ∞] sn
 = lim[n → ∞] (306n^2 + 35n + 306)
 = ∞
So unfortunately, the series (n + 16)(n + 18) diverges to infinity and does not have a finite sum.To find the sum S of the series (n + 16)(n + 18), we need to take the limit of the sequence of partial sums as n approaches infinity. So let's first find the formula for the nth partial sum sn:

sn = (1 + 16)(1 + 18) + (2 + 16)(2 + 18) + ... + (n + 16)(n + 18)
  = ∑[(k + 16)(k + 18)] (from k = 1 to n)

Using the formula for the sum of squares, we can expand each term in the sum:

(k + 16)(k + 18) = k^2 + 34k + 288

So now we have:

sn = ∑(k^2 + 34k + 288) (from k = 1 to n)
  = ∑k^2 + 34∑k + 288n (from k = 1 to n)
  = n(n + 1)(2n + 1)/6 + 34n(n + 1)/2 + 288n
  = 306n^2 + 35n + 306

Now we can take the limit of sn as n approaches infinity to find S:

S = lim[n → ∞] sn
 = lim[n → ∞] (306n^2 + 35n + 306)
 = ∞

So unfortunately, the series (n + 16)(n + 18) diverges to infinity and does not have a finite sum.

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If she continues to increase the length of her runs the same way, it will not be enough to reach the half-marathon distance of 13.1 miles within the remaining time of the training program.

Sariah has just begun training for a half-marathon, which is 13.1 miles. There are 6 weeks of training runs remaining before the race. In her first week of training, Sariah ran 3 miles. She ran 4.5 miles the second week and 6 miles the third week.

To determine if there is enough time left in the training program for her to get up to half-marathon distance, let's analyze the pattern of her weekly increases in distance:

Week 2 - Week 1 = 4.5 miles - 3 miles = 1.5 miles increase
Week 3 - Week 2 = 6 miles - 4.5 miles = 1.5 miles increase

Sariah is consistently increasing her weekly mileage by 1.5 miles. With 3 weeks of training already completed, she has 3 more weeks to go. Let's see if she can reach the half-marathon distance of 13.1 miles:

Week 4: 6 miles + 1.5 miles = 7.5 miles
Week 5: 7.5 miles + 1.5 miles = 9 miles
Week 6: 9 miles + 1.5 miles = 10.5 miles

After 6 weeks of training, Sariah will have increased her longest run to 10.5 miles. Unfortunately, this is not enough to reach the half-marathon distance of 13.1 miles within the remaining time of the training program.

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The mode of transportation that I will choose is motorized scooter this is because it is relatively cheap when compared to the other forms of transportation.

A motorized scooter  could be a moderately cheap mode of transportation compared to other alternatives such as cars, cruisers, or indeed bikes.

The starting price of obtaining a motorized bike is for the most part lower than that of a car or cruiser, and the continuous costs such as fuel, support, and protections are too ordinarily lower.

From the question, you can see that the price of the scooter at 5 miles is $10 and it is relatively cheap when compared to the rest.

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The grocery store's earning income is $30,030 if they sell 200 tomatoes in one day.

We need to find how much the grocery store earns when it sells 200 tomatoes in one day. When The grocery store’s earnings in dollars can be modeled by the equation,

y = 0.75x² + 0.15x

where,

x = number of tomatoes they sell = 200

To find the earnings we need to substitute x in the equation it can be given as,

y = 0.75x² + 0.15x

y = 0.75(200)² + 0.15(200)

y = $30,030

Therefore, the grocery store's earning income is $30,030 if they sell 200 tomatoes in one day.

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Step-by-step explanation:

To determine the appropriate fraction for the line plot, we need to find the greatest common factor (GCF) of all the pencil lengths, and then express each length as an equivalent fraction with the GCF as the denominator.

The GCF of the pencil lengths is 1. Therefore, we can simply express each pencil length as an equivalent fraction with 1 as the denominator:

3/1, 7/1, 8/1, 5/1, 1/1, 4/1, 6/1, 8/1, 4/1, 2/1, 5/1

Now, we can see that the smallest unit increment we can use on the line plot is 1/8. This is because 1/8 is the smallest fraction that can represent all of the pencil lengths above the label (5/8, which is equivalent to 10/16).

Therefore, the students should use 1/8 as the fraction for the line plot.

The value of the given expression is 44. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

The four basic operations, also referred to as "arithmetic operations," are thought to explain all real numbers. Operations like division, multiplication, addition, and subtraction come before operations like quotient, product, sum, and difference in mathematics.

We are given an expression as [tex]\frac{1}{2}[/tex] (7) (4) + 6 (5).

We know that when there is no sign in between two numbers, it denotes multiplication.

So, we get

⇒  [tex]\frac{1}{2}[/tex] * (7)  * (4) + 6 * (5)

⇒ 14 + 30

⇒ 44                                            (Using addition operation)

Hence, the value of the given expression is 44.

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Answer:

(3-x)^2=y

Step-by-step explanation:

The cyclist needs to ride approximately 23.21 km in the direction of 23.13° W of North to get back to the starting point.

We can use trigonometry to find the x and y components.

x3 = 12 * cos(15°) ≈ 11.59 km (east)

y3 = -12 * sin(15°) ≈ -3.10 km (south, hence the negative sign)

Leg #4: 14 km [South]

x4 = 0 km (no east/west movement)

y4 = -14 km (south, hence the negative sign)

Now, let's find the total x and y displacements:

x_total = x1 + x2 + x3 + x4 ≈ 0 + 10 + 11.59 + 0 ≈ 21.59 km

y_total = y1 + y2 + y3 + y4 ≈ 8 + 0 - 3.10 - 14 ≈ -9.10 km

Now, we can find the distance and direction he needs to ride to get back to the starting point:

Distance=  

[tex]\sqrt{21.59^2 + (-9.10)^2}[/tex])

23.21 km

Direction:

angle =

arctan(9.10 / 21.59)

= 23.13°

The cyclist needs to ride approximately 23.21 km in the direction of 23.13° W of North to get back to the starting point.

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The function f representing the sum of the series for x in the interval (-3, 3). Hi! The given geometric series is 1 - x/3 - x^2/9 - x^3/27...

The common ratio of the series is obtained by dividing a term by its preceding term. Let's consider the first two terms:

(-x/3) / 1 = -x/3

Therefore, the common ratio (r) of the series is -x/3.

For a geometric series to converge, the absolute value of the common ratio must be less than 1, i.e., |r| < 1. In this case:

|-x/3| < 1

To find the values of x for which the series converges, we need to solve the inequality:

-1 < x/3 < 1

Multiplying all sides by 3, we get:

-3 < x < 3

So, the series converges for x in the interval (-3, 3).

Now, let's determine the function f representing the sum of the series. For a converging geometric series, the sum S can be calculated using the formula:

S = a / (1 - r)

where a is the first term and r is the common ratio. In this case, a = 1 and r = -x/3. Therefore:

f(x) = 1 / (1 - (-x/3))
f(x) = 1 / (1 + x/3)

This is the function f representing the sum of the series for x in the interval (-3, 3).

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