Use the Intermediate Value Theorem to identify the location of the first positive root in f(x)=x²-3

Continuous-compounding is a method of calculating interest where the interest is added to the principal continuously.

instead of being added at regular intervals (such as monthly or annually). This means that the interest is compounded an infinite number of times-over the year, resulting in a higher effective interest rate than other compounding methods.

In this scenario, the person has invested [tex]$479[/tex] in an account that earns an annual interest rate of [tex]8.2%[/tex] compounded continuously. This means that the interest is added to the account balance continuously throughout the year.

The formula for calculating the balance of an account with continuous compounding is:

[tex]V = Pe^(rt)[/tex]

where:

V = the balance after t years

P = the initial investment (or principal)

e = the mathematical constant approximately equal to [tex]2.71828[/tex]

r = the annual interest rate as a decimal

t = the number of years

Using this formula and substituting the given values, we get:

[tex]V = 479e^(0.08212)[/tex]

Simplifying this expression, we get:

[tex]V ≈ $1,204.70[/tex]

Therefore, the person's investment of [tex]$479[/tex] with an annual interest rate of [tex]8.2%[/tex]  compounded continuously, would grow to approximately after 12 years

The formula for calculating the value of the account after t years, with continuous compounding, is:

[tex]V = Pe^(rt)[/tex]

where V is the final value, P is the initial principal, r is the interest rate (expressed as a decimal), and t is the time in years.

Using this formula, we can calculate the value of the account after 12 years:

[tex]V = 479 * 2.6709[/tex]

[tex]V = 1280.74[/tex]

Final answer

Therefore, the amount of money in the account after [tex]12[/tex] years, to the nearest cent, is [tex]$1,280.74.[/tex]

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The value of the stock decreased by $203.

We have,

The initial value of the stock is:

= 25 shares X $35.48/share

= $887

The current value of the stock is:

= 25 shares x $27.36/share

= $684

The value of the stock decreased by:

= $887 - $684

= $203

Thus,

The value of the stock decreased by $203.

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(1) The two triangles are similar because they have equal angles.

(2)  Triangle QRS is similar to triangle QLM because they have equal angles.

(3) Both triangles are similar and the value of x is 21.

Two triangles are said to be similar if they have equal sides, equal angles or both.

The missing angles of the triangles for the question is calculated as;

Bigger triangle; missing angle = 180 - (44 + 46) = 90

Smaller triangle; missing angle = 90 - 46 = 44⁰

Both triangles are similar.

For the second question; triangle QRS is similar to triangle QLM  because angle R is equal to angle L, and also they have common angle Q, which implies that angle S must be equal to angle L.

For third question, the triangles are similar because their corresponding angles are equal.

The value of x is calculated as;

48 + 4x + (180 - (56 + 76)) = 180 (sum of angles on a straight line)

48 + 4x + 48 = 180

4x = 84

x = 84/4

x = 21

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Answer:

We can use the formula for the probability mass function of a binomial distribution:

P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)

Where:

n = number of trials

k = number of successes

p = probability of success

In this case, n = 5, k = 1, and p = 0.05. Plugging these values into the formula, we get:

P(X = 1) = (5 choose 1) * 0.05^1 * (1 - 0.05)^(5-1) ≈ 0.23

Rounding to the nearest tenth, the probability of exactly one success in five trials with a 5% probability of success is approximately 0.2 or 20%.

Step-by-step explanation:

The upper limit of the control chart such that it will include roughly 97 percent of the sample means when the process is in control is 1.008.

For a process with a normal distribution, the mean of the sample means is equal to the population mean and the standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size. In this case, the mean of the output is 1.0 liter and the standard deviation is 0.01 liter, so the standard deviation of the sample means is 0.01 / √25 = 0.002.

To construct a control chart for the sample means, we need to determine the upper and lower control limits such that the process is in control when the sample means fall within these limits. Assuming the process is in control, we want to find the upper limit such that roughly 97% of the sample means fall below this limit.

Using the standard normal distribution, the Z-score corresponding to the 97th percentile is approximately 1.88.

Therefore, the upper control limit is 1.0 + 1.88(0.002) = 1.008. Any sample mean above this limit should be investigated for potential process issues.

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Answer:

74% is about 75%, or 3/4.

57 is about 60.

So 74% of 57 is about 3/4 of 60, or 45.

74% of 57 is .74 × 57 = 42.18, so the estimate seems reasonable

Answer:

42.18

Step-by-step explanation:

[tex]\frac{57*74}{100} = 42.18[/tex]

The equations of the asymptotes for the graph of F(x) are Vertical asymptote at x = 4 and Horizontal asymptote at y = 0.

To find the equations of the asymptotes, we need to examine the behavior of the function as x gets very large or very small.

First, let's factor the denominator of the function

F(x) = (x + 3) / (x - 4)(x + 3)

Notice that (x + 3) appears in both the numerator and denominator, and therefore can be cancelled out, leaving

F(x) = 1 / (x - 4)

Now, as x gets very large or very small, the value of F(x) approaches 0. However, we can see that as x approaches 4, the denominator of F(x) approaches 0, which means F(x) approaches infinity or negative infinity, depending on which side of x = 4 we approach from.

Therefore, we have a vertical asymptote at x = 4.

To find any horizontal asymptotes, we need to examine the behavior of the function as x approaches infinity or negative infinity. Since the degree of the numerator and denominator are the same (both 1), we can find the horizontal asymptotes by looking at the ratio of the leading coefficients

F(x) = (x + 3) / (x - 4)(x + 3)

As x approaches infinity or negative infinity, the denominator becomes dominated by the highest degree term, x². Therefore

F(x) ≈ (1/x²) / (1 - 4/x + 3/x²)

As x approaches infinity or negative infinity, the terms with x in the denominator become negligible compared to the constant term. Therefore

F(x) ≈ (1/x²) / (1 + 0 + 0) = 1/x²

Thus, we have a horizontal asymptote at y = 0.

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The probability that a randomly chosen voter from the survey  is under 50 years old is 0.75

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

The table of values

Where we have

Voters under 50 years old = 847 + 804 + 773

Total = 3228

So, the required probability is

P = (847 + 804 + 773)/3228

Evaluate

P = 0.75

Hence, the probability is 0.75

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The value of x in the rectangular prism is 9 inches.

The height of the rectangular prism can be found as follows:

The volume of the rectangular prism is 153 inches cube.

Therefore,

volume of the rectangular prism = lwh

where

Therefore,

volume of the rectangular prism = 8.5 × 2 × x

153 = 17x

divide both sides by 17

x = 153 / 17

x = 9 inches

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The value of the test statistics is 113.42.

To test the claim that the standard deviation of pulse rates of adult males is less than 10 bpm, we will use a chi-square test. Given a random sample of 135 adult males with a standard deviation of 9.2 bpm, let's find the value of the test statistic.

Step 1: State the null and alternative hypotheses.
H0: σ = 10 bpm (null hypothesis)
H1: σ < 10 bpm (alternative hypothesis)

Step 2: Determine the appropriate test statistic.
In this case, we will use the chi-square test statistic: χ² = (n-1)(s²/σ²), where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation.

Step 3: Calculate the test statistic.
χ² = (135-1)(9.2²/10²) = (134)(84.64/100) = 134 * 0.8464 ≈ 113.42

The value of the test statistic is approximately 113.42. This value can be compared to the critical value from the chi-square distribution table with (n-1) degrees of freedom to determine whether to reject or fail to reject the null hypothesis.

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The break-even x-value for Year-round-Recreation is approximately 26.19.

To find the break-even points for Year-round-Recreation, we need to set the total costs equal to total revenues and solve for x. In this case, the total costs are given by C(x) = 2750 + 30x + 2, and the total revenues are given by R(x) = 135x.

The break-even point is when C(x) = R(x), so:

2750 + 30x + 2 = 135x

Now, we need to solve for x:

1. Subtract 30x from both sides:
2750 + 2 = 105x

2. Subtract 2 from both sides:
2750 = 105x

3. Divide both sides by 105:
x = 2750 / 105
x ≈ 26.19

The break-even x-value for Year-round-Recreation is approximately 26.19.

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The student solved 12 problems correctly.

To find out how many problems the student solved correctly, we can use the given information and set up an equation using the terms "correct problems" and "incorrect problems."

Let x represent the number of correct problems and y represent the number of incorrect problems. We know the following:

1. The total number of problems completed is 33, so x + y = 33.
2. The teacher pays 70 cents for correct problems and takes 40 cents for incorrect problems, and neither owes anything to each other. So, 70x - 40y = 0.

Now, we'll solve this system of equations step-by-step:

Step 1: Solve the first equation for x: x = 33 - y.
Step 2: Substitute the expression for x in the second equation: 70(33 - y) - 40y = 0.
Step 3: Simplify and solve for y: 2310 - 70y - 40y = 0 => 2310 - 110y = 0 => y = 21.
Step 4: Substitute the value of y back into the equation for x: x = 33 - 21 => x = 12.

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Answer:

The answer to your problem is, 63

Step-by-step explanation:

So we know that he has 2,835 comic books. He is also going to put them in boxes to ship it in a book store.

1 Box = 45 Comic Books

So in order to solve the problem we need to divide:

The expression includes:

2,835 ÷ 45 = 63

Thus the answer to your problem is, 63

A line starts at 0 with a constant positive slope.

The graph that shows the distance Sam travels over time is given below.

Here the graph starts at the point (0, 0).

So the line starts at 0.

Now the line is moving in such a way that as time increases, the distance travelled also increases.

So the slope is positive.

Hence the complete description is that line starts at 0 with a constant positive slope.

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The final value is 4 (mod5)

Hence, 17^342 ≡ 2^342 ≡ 4 (mod5).

To find 5^903 (mod60), we can use Euler's totient function. Since 60 = 2^2 × 3 × 5, we have φ(60) = 2^1 × 3^1 × 4 = 24. Therefore, we can use Euler's theorem to write:

5^24 ≡ 1 (mod60)

Raising both sides to the power of 37, we get:

5^(24*37) ≡ 1^37 ≡ 1 (mod60)

So 5^888 ≡ 1 (mod60).

Now, we can write:

5^903 = 5^888 * 5^15

Since 5^888 ≡ 1 (mod60), we just need to find 5^15 (mod60).

To do this, we can use the repeated squaring method. Writing 15 in binary form, we have:

15 = 1111 (in binary)

So we can compute:

5^1 ≡ 5 (mod60)

5^2 ≡ 25 (mod60)

5^4 ≡ 25^2 ≡ 25 (mod60)

5^8 ≡ 25^2 ≡ 25 (mod60)

Therefore:

5^15 ≡ 5^8 * 5^4 * 5^2 * 5^1 ≡ 25 * 25 * 25 * 5 ≡ 25 (mod60)

Hence, 5^903 ≡ 5^15 ≡ 25 (mod60).

To find 17^342 (mod5), we can use the fact that 17 ≡ 2 (mod5). Therefore:

17^342 ≡ 2^342 (mod5)

Using the repeated squaring method again, we can compute:

2^1 ≡ 2 (mod5)

2^2 ≡ 4 (mod5)

2^4 ≡ 1 (mod5)

Therefore:

2^342 ≡ 2^2 * (2^4)^85 ≡ 4 * 1^85 ≡ 4 (mod5)

Hence, 17^342 ≡ 2^342 ≡ 4 (mod5).

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The number or expression underneath the top line of the symbol is called the radicand. The cube root symbol is a grouping symbol, meaning that all operations in the radicand are grouped as if they were in parentheses.

To write the radicand as the product of a perfect cube (first) and a factor that does not contain a perfect cube (second), follow these steps:

1. Identify the radicand in the given expression. The radicand is the number or expression inside the cube root symbol.
2. Determine the prime factors of the radicand by breaking it down into its smallest prime factors.
3. Group the prime factors into sets of three identical factors. These sets will form the perfect cube factors.
4. Multiply the sets of three factors together to form the perfect cube part of the product.
5. Multiply any remaining factors together to form the factor that does not contain a perfect cube.
6. Write the radicand as the product of the perfect cube (step 4 result) and the factor that does not contain a perfect cube (step 5 result).

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Answer:

The correct answer is option B: 2x^2 + 11x + 12 cubic inches

Step-by-step explanation:

The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width (or base), and h is the height of the prism.

Given that the area of the base is (x + 4) square inches and the height is (2x + 3) inches, we can substitute these values into the formula to find the volume:

V = (x + 4)(2x + 3)

Now, we can multiply the binomials using the distributive property:

V = 2x^2 + 3x + 8x + 12

V = 2x^2 + 11x + 12

So, the correct answer is option B: 2x^2 + 11x + 12 cubic inches.

The 7th term of the sequence is 0.297

We have,

To find the 7th term, we need to know the common ratio.

We can find the common ratio by dividing any term by the previous term.

Common ratio = 6/18 = 1/3

Now we can use the formula for the nth term of a geometric sequence:

[tex]a_n = a_1 \times r^{n-1}[/tex]

where a(1) is the first term and r is the common ratio.

So, for this sequence:

a(1) = 18

r = 1/3

To find the 7th term:

a(7) = 18 x (1/3)^{7 - 1}

      = 0.297

Rounding to the nearest thousandth:

Thus,

The 7th term of the sequence is 0.297

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Answer:

In the given diagram, QP is tangent to a circle with centre O, RS is a straight line, T is a point on the circle, and PS bisects TPQ. We know that SPQ = 22°. Let's try to find out the value of the angle TPQ.

Since QP is tangent to the circle, the angle between RS and QP (i.e., angle RQP) is equal to the angle between QP and the radius drawn to the point of tangency (i.e., angle QOT). So, we can say that:

angle RQP = angle QOT

Also, since PS bisects TPQ, we can say that:

angle TPS = angle TPQ / 2

Now, let's consider the triangle TPQ. We know that:

angle TPQ + angle TQP + angle PTQ = 180° [Sum of angles in a triangle]

Substituting the values we have:

angle TPQ + angle TQP + (angle TPS + angle SPQ) = 180°

angle TPQ + angle TQP + (angle TPQ/2 + 22°) = 180°

Multiplying both sides by 2 to eliminate the fraction:

2(angle TPQ) + 2(angle TQP) + angle TPQ + 44° = 360°

Simplifying:

3(angle TPQ) + 2(angle TQP) = 316°

We don't know the values of angle TPQ and angle TQP, so we can't solve this equation exactly. However, we do know that these angles are both less than 180° (since they are angles in a triangle). Therefore, we can try some values for angle TPQ (let's call it x) and see if we can find a corresponding value for angle TQP that satisfies the equation.

If we take x = 40°, then we get:

3(40°) + 2(angle TQP) = 316°

120° + 2(angle TQP) = 316°

2(angle TQP) = 196°

angle TQP = 98°

Now, we can use the fact that angle TPS = angle TPQ / 2 to find angle TPS:

angle TPS = x/2 = 20°

Finally, we can use the fact that PS bisects TPQ to find angle PQT:

angle PQT = angle TPS

Step-by-step explanation:

The given  statement pattern is a tautology.

To determine whether the following statement pattern is a tautology, a contradiction, or a contingency: (p→q)∨(q→p), follow these steps:

1. Write out the truth table for p and q:
  p | q
  -----
  T | T
  T | F
  F | T
  F | F

2. Calculate the truth values for (p→q) and (q→p) using the implication rule (p→q is false only when p is true and q is false):
  (p→q) | (q→p)
  -----------
   T     |   T
   F     |   T
   T     |   F
   T     |   T

3. Finally, calculate the truth values for the given statement pattern (p→q)∨(q→p) using the disjunction rule (a disjunction is true if at least one of the statements is true):
  (p→q)∨(q→p)
  ---------
     T
     T
     T
     T

Since the statement pattern (p→q)∨(q→p) is true for all possible truth values of p and q, it is a tautology.

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The appropriate conclusion at a 5% significance level is that a new employee needs to be hired since the p-value is less than 0.05.

To test the hypothesis, we will use a one-sample t-test with a null hypothesis that the true population mean wait time is less than or equal to 5 minutes. The alternative hypothesis is that the true population mean wait time is greater than 5 minutes.

Using the given sample data, we calculate the sample mean (x-bar) as 5.53 and the sample standard deviation (s) as 0.67. The sample size is 7.

We calculate the t-statistic using the formula t = (x-bar - mu)/(s/sqrt(n)), where mu is the hypothesized population mean (5) and n is the sample size.

Substituting the values, we get t = (5.53 - 5)/(0.67/sqrt(7)) = 2.44.

Using a t-distribution table with 6 degrees of freedom (n-1), we find the p-value to be 0.03 for a one-tailed test. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that a new employee needs to be hired to reduce the average wait time.

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The area of the pizza box measuring 2/3 feet wide by 4/5 feet long is 8/15 square feet.

The shape of the pizza box is a rectangle. The rectangle is a quadrilateral with opposite sides parallel and equal with an equal angle and of 90°.

The area of a rectangle is considered as:

A = L * B

where L is the length

B is the breadth

Given in the question,

L = 4/5 feet

B = 2/3 feet

The area is calculated by multiplying the fractions. For the multiplication of fractions, we multiply the numerators and denominators separately. And final answer is calculated by simplifying the resulting fraction.

A = 4/5 * 2/3

= 4*2 / 5*3

= 8/15 square feet

Thus, the pizza box has an area of 8/15 square feet

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The amount of space within the cylinder not taken up by the tennis balls is 18.9 [tex]inches^3[/tex]

The volume of a tennis ball:

The circumference of the tennis ball is 8 inches.

The tennis ball is the form of sphere whose circumference is given by formula [tex]2\pi r[/tex] , where r is the radius.

Thus, if r is the radius then according to condition,

[tex]2\pi r[/tex]  = 8 or

r = 8/2[tex]\pi[/tex] inches.

Now, the volume of the sphere of radius r is [tex]\frac{4}{3}\pi r^3[/tex] hence, find the volume of the given tennis ball by substituting r  = 8/2[tex]\pi[/tex] inches in [tex]\frac{4}{3}\pi r^3[/tex] and simplify:

Volume = [tex]\frac{4}{3} \pi[/tex] × [tex](\frac{8}{2\pi } )^3[/tex]

Volume = 8.65 [tex]inches^3[/tex]

Hence the required volume of the tennis ball is 8.65 [tex]inches^3[/tex]

The volume of three tennis balls is (3 × 8.65) [tex]inches^3[/tex]  = 25.96 [tex]inches^3[/tex]

Find the volume of the cylinder:

The volume of the cylinder with radius r units and height h units is given by  [tex]\pi r^2h[/tex] Hence the volume of the given cylinder with radius 1.32 inches , 8.2 height inches is:

[tex]\pi (1.32)^2[/tex] × 8.2

= 3.14 × [tex](1.32)^2[/tex] × 8.2

= 44.86 [tex]inches^3[/tex]

Hence the volume of the cylinder is 44.86 [tex]inches^3[/tex]

Find the amount of space within the cylinder not taken up by the tennis balls.

The required volume can be obtained by subtracting the volume three tennis balls from the volume of the cylinder as follows:

Volume of cylinder - volume of three tennis balls = (44.86 - 25.96) = 18.9 [tex]inches^3[/tex]

Hence, the amount of space within the cylinder not taken up by the tennis balls is 18.9 [tex]inches^3[/tex].

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Answer:

x = 5√3, and y = 10.

In a 30°-60°-90° triangle, the length of the longer leg is √3 times the length of the shorter leg, and the length of the hypotenuse is twice the length of the shorter leg.

a) The equilibrium point is x = 2 or x = 8

b) The consumer surplus equilibrium point is $6,062.67

c) The producer surplus equilibrium point is $13,208.67

a) To find the equilibrium point, we need to set the demand function equal to the supply function and solve for x:

D(x) = S(x)

[tex](x - 5)^2 = x^2 + x + 3[/tex]

Expanding the left side and simplifying, we get:

[tex]x^2 - 10x + 22 = 0[/tex]

Using the quadratic formula, we get:

[tex]x = (10[/tex] ± [tex]\sqrt{36})/ 2[/tex]

[tex]x = 5[/tex] ± [tex]3[/tex]

[tex]x = 2[/tex] or [tex]x = 8.[/tex]

b) To find the consumer surplus at the equilibrium point, we need to calculate the area under the demand curve and above the equilibrium price, which is given by the supply curve. Since we have two possible equilibrium points, we need to check both of them to see which one gives us a positive consumer surplus.

For [tex]x = 2[/tex], the equilibrium price is given by [tex]S(2) = 11[/tex], which is above the demand curve. Therefore, there is no consumer surplus at this equilibrium point.

For [tex]x = 8[/tex], the equilibrium price is given by [tex]S(8) = 75[/tex], which is below the demand curve. Therefore, the consumer surplus is given by the area under the demand curve and above the price of 75:

[tex]∫[75, 8] (x - 5)^2 dx = [(x - 5)^3 / 3][/tex] from 8 to 75

≈[tex]6,062.67[/tex]

This means that at the equilibrium point x = 8, consumers are willing to pay a total of approximately $6,062.67 more than what they actually pay.

c) To find the producer surplus at the equilibrium point, we need to calculate the area under the equilibrium price and above the su

For x =2 supply curve. Again, since we have two possible equilibrium points, we need to check both of them to see which one gives us a positive producer surplus.

2, the equilibrium price is given by [tex]S(2) = 11,[/tex] which is above the demand curve. Therefore, there is no producer surplus at this equilibrium point.

For x = 8, the equilibrium price is given by[tex]S(8) = 75[/tex], which is below the demand curve. Therefore, the producer surplus is given by the area above the supply curve and below the price of 75:

∫[tex][8, 75] (75 - x^2 - x - 3) dx = [(75x - x^3/ 3 - x^2 / 2 - 3x)][/tex] from 8 to 75

≈ [tex]13,208.67[/tex]

This means that at the equilibrium point x = 8, producers receive a total of approximately $13,208.67 more than their costs.

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Estimating the square root of the number -√7 gives -2 and -2.6

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

-√7

To estimate the number is to approximate the number

When the square root of 7 is evaluated, we have

-√7 = -2.64575131106

Approximate to the nearest integer

-√7 = -2

Approximate to the nearest tenth.

-√7 = -2.6

Hence, the estimates are -2 and -2.6

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The word "sleepless" has 8 letters. To find the number of distinguishable permutations, we can use the formula for permutations of a set with no repeated elements, which is n!, where n is the number of elements.

Therefore, the number of permutations for the word "sleepless" can be calculated as 8!, which is equal to 40,320. This means that there are 40,320 different ways we can arrange the letters in the word "sleepless" while keeping all the letters distinct.

Note that if the word had repeated letters, we would have to divide the result by the factorials of the number of times each letter was repeated.

The unknown angle of the parallelogram is as follows:

m∠T = 63 degrees

A parallelogram is a quadrilateral with opposite sides parallel to each other and opposite congruent to each other.

Therefore, the opposite angles of a parallelogram are equal. Consecutive angles are supplementary angles to each other.

Hence,

10w + 53 + 17w + 100 = 180

27w + 153 = 180

27w = 180 - 153

27w = 27

divide both sides by 27

w = 27 / 27

w = 1

Therefore,

m∠T = 10(1) + 53 = 63 degrees

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Answer:

-3x^2 + 12x + 5x - 20 = -3x^2 + 17x - 20

Step-by-step explanation:

(-3x - 5)(-x + 4) is a binomial expression, where (-3x - 5) is one expression and (-x + 4) is the other.

As the text eludes to, we can multiply binomial expressions using the FOIL method, where you multiply the first terms (3x and -x), outer terms (3x and 4), the inner terms (-5 and -x), and the last terms (-5 and 4)

This is how you get

(3x)(-x) + (3x)(4) + (-5)(-x) + (-5)(4)

Now, multiply the terms and combine like terms:

[tex](3x)(-x)+(3x)(4)+(-5)(-x)+(-5)(4)\\-3x^2+12x+5x-20\\-3x^2+17x-20[/tex]