Find the 29th term of the following sequence. 19.5, 19.9, 20.3, 20.7, . . .

The correct option is B) [tex]x=4.4[/tex]. The value of x to the nearest tenth is x=4.4.

To find the value of x, we need to apply the Law of Cosines to the given triangle. The Law of Cosines states that for any triangle with sides [tex]\( a \), \( b \), and \( c \),[/tex] and the angle opposite side c and [tex]\( \gamma \),[/tex] the following relationship holds: [tex]\[ c^2 = a^2 + b^2 - 2ab\cos(\gamma) \][/tex]

Let's denote the sides of the triangle as follows: the side opposite the right angle (hypotenuse) as c, the side opposite the 30-degree angle as b, and the side opposite the 60-degree angle as a. We are given that [tex]\( a = 2x \), \( b = x \), and \( c = 10 \).[/tex]

Using the Pythagorean theorem, we have:

[tex]\[ c^2 = a^2 + b^2 \][/tex]

[tex]\[ 10^2 = (2x)^2 + x^2 \][/tex]

[tex]\[ 100 = 4x^2 + x^2 \][/tex]

[tex]\[ 100 = 5x^2 \][/tex]

[tex]\[ x^2 = \frac{100}{5} \][/tex]

[tex]\[ x^2 = 20 \][/tex]

[tex]\[ x = \sqrt{20} \][/tex]

[tex]\[ x = \sqrt{4 \cdot 5} \][/tex]

[tex]\[ x = 2\sqrt{5} \][/tex]

Now, we calculate the numerical value of x:

[tex]\[ x \approx 2 \cdot 2.236 \][/tex]

[tex]\[ x \approx 4.472 \][/tex]

Rounding to the nearest tenth, we get:  [tex]\[ x \approx 4.4 \][/tex]

Therefore, the value of x to the nearest tenth is 4.4, which corresponds to option B.

Answer:

A. Barry spends $9 on lunch daily.

Step-by-step explanation:

We are required,

To find the situation in which one constant is changing at a constant rate in relation to the other quantity.

So, according to the options, we have,

A. Barry spends $9 on lunch daily.

Let x= number of days and as the cost of lunch each day is $9

Then, the total cost of the lunch y, is [tex]y=9x[/tex]

Thus, this situation changes the variable with constant rate.

B. The market value of a new computer decreases by 6% every year.

Let, a= fixed initial value of the computer and x= number of years.

Since, the rate of decrease in value = 6% = 0.06

Then, the market value of the computer per year, y, is given by,

[tex]y=a(1-0.06)^x\\\\y=a(0.94)^x[/tex]

Thus, this situation changes the variable exponentially.

C. Nathan's investments increase in value by 13.5% each month.  

Let, a= fixed initial investment and x= number of months.

Since, the rate of increase = 13.5% = 0.135

Then, the investment per month, y, is given by,

[tex]y=a(1+0.135)^x\\\\y=a(1.135)^x[/tex]

Thus, this situation changes the variable exponentially.

D. The number of ants in a colony doubles every 8 weeks.

Let, x= number of weeks where 'x' belongs to the se {1,8,16,24,...}.

Since, the number of ants is doubling every 8 weeks.

So, the number of ants in the colony, y, is [tex]y=2^x[/tex]

Thus, this situation changes the variable exponentially.

QUESTION 1

The area of a triangle is given by

[tex]Area=\frac{1}{2}\times base\times heigth[/tex]

The length of the base of the triangle is [tex]6\:in.[/tex] and the vertical height is [tex]8\:in.[/tex].

We substitute these values into the formula to obtain,

[tex]Area=\frac{1}{2}\times 6\times 8[/tex]

[tex]Area=3\times 8[/tex]

[tex]Area=24\:in^2[/tex]

The area of the triangle is 24 square inches.

The correct answer is A

QUESTION 2

The company logo is made up of three triangles and a square.

The length of the square is [tex]7\:cm[/tex]

The area of a square is given by

[tex]Area=l^2[/tex]

[tex]\Rightarrow Area=7^2[/tex]

[tex]\Rightarrow Area=49cm^2[/tex].

The area of one of the triangles is

[tex]Area=\frac{1}{2}\times base \times height[/tex]

The height of the triangle is [tex]4cm[/tex].

The base of the triangle is on one side of the square, so it is [tex]7cm[/tex].

The area now becomes

[tex]Area=\frac{1}{2}\times 7 \times 4[/tex]

[tex]\Rightarrow Area=7 \times 2[/tex]

[tex]\Rightarrow Area=14cm^2[/tex].

Since there are three identical triangles, we multiply the area of one triangle by 3 to get area of the three triangles.

[tex]Area\:of\:the\:three\:triangles=3\times14=42cm^2[/tex]

The area of the logo is equal to the area of the square plus the area of the three identical triangles.

[tex]Area\:of\:logo=49+42=91cm^2[/tex]

Hence the area of the logo is [tex]91cm^2[/tex]

QUESTION 3

Since the figure is made up of two rectangles and two right triangles, we find their areas and sum them to get the area of the figure.

The area of a rectangle is given by

[tex]Area=l\times w[/tex]

The width of the bigger rectangle is [tex]5[/tex] and the length is [tex]25[/tex].

[tex]Area\:of\:bigger\: rectangle=25\times 5[/tex]

[tex]Area\:of\:bigger\: rectangle=125\:square\:units[/tex]

The width of the smaller rectangle is [tex]8[/tex].

The length of the smaller rectangle is [tex]25-(6+6)=25-12=13[/tex].

[tex]Area\:of\:smaller\: rectangle=13\times 8[/tex]

[tex]Area\:of\:smaller\: rectangle=104\:square\:units[/tex].

The two triangles are identical, so we find the area of one and multiply by 2

[tex]Area\:of\:triangle=\frac{1}{2}\times base \times heigth[/tex]

[tex]Area\:of\:triangle=\frac{1}{2}\times 6 \times 8[/tex]

[tex]Area\:of\:triangle=3 \times 8[/tex]

[tex]Area\:of\:triangle=24\:square\:units[/tex]

[tex]\Rightarrow Area\:of\:the\:two\:triangles=2\times24=48\:square\:units[/tex]

The area of the figure is

[tex]=125+104+48=277\:sqaure\:units[/tex]

The correct answer is C

The linearization L(x) of the function y=e8xln(x) at a=1, is given by L(x) = f(a) + f'(a)(x - a). Following the steps of calculating the function at a=1, determining its derivative and evaluating this at a=1, then substituting these into the linearization formula, this comes out to be L(x) = 1 + 8(x - 1).

The linearization of a function at a point is approximately equivalent to the function's tangent line at that point. The formula for linearization, L(x), can be given as L(x) = f(a) + f'(a)(x - a). In this case, find the linearization L(x) of y=e8xln(x) at a=1. To do this, we'll follow a step-by-step process:

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After the multiplication of the given fraction, the value obtained in terms of a fraction will be equal to 35/9.

In mathematics, a fraction is represented by a specific number that designates a portion of an entire. A fraction is a piece or section of a whole, whereby a can have been any number, a particular value, or an object.

As per the information available in the question,

The given fraction is,

1 2/3 × 2 1/3

It can also be written as:

5/3 × 7/3 = 35/9 or 3 8/9.  

To know more about Fraction:

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Answer:

1 and 3/2 or 1 and 1 1/2

Step-by-step explanation:

3➗3=1 and 11➗11= 1 so a. is 1/1 or 1.

9/10➗3/5 or 9➗3=3 and 10➗5=2 so it is 3/2 or 1 1/2 so the answers are 1 and 1 1/2! Hope this helps.

The equivalent factored form of 12x4 - 42x3 - 90x2 is 6x(x - 5)(2x + 3).

The equivalent factored form of 12x4 - 42x3 - 90x2 is 6x(x - 5)(2x + 3).

We can factor out the greatest common factor, which is 6x. That leaves us with 2x3 - 7x2 - 15x.

Then we can factor by grouping, which involves grouping terms that have a common factor. By factoring out an x from the first two terms, we get x(2x2 - 7x - 15). By factoring out a -5 from the last two terms, we get -5(2x2 - 7x - 15). This gives us the final factored form of 6x(x - 5)(2x + 3).

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To solve the expression 1/7 - 3(3/7n - 2/7), distribute the 3 to the terms inside the parentheses, simplify the expression inside the parentheses, and combine like terms to get the simplified form of 7n - 1/7.

To solve the expression 1/7 - 3(3/7n - 2/7), we need to follow the order of operations which is Parentheses, Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).

First, distribute the 3 to the terms inside the parentheses: 1/7 - (9/7n - 6/7).

Next, simplify the expression inside the parentheses: 1/7 - 9/7n + 6/7.

Finally, combine like terms by subtracting and adding the fractions: 7n - 1/7.

So, the simplified form of the expression 1/7 - 3(3/7n - 2/7) is 7n - 1/7.

Final answer:

The algebraic expression 1/7 - 3(3/7n - 2/7) can be simplified by distributing the -3 and combining like terms to yield a final expression of 1 - 9/7n.

Explanation:

The student is asking for help in solving an algebraic expression. The problem given is 1/7 - 3(3/7n - 2/7). To solve this, we need to apply the distributive property and then simplify the expression. Here's how we can solve it step by step:

Distribute the -3 to the terms inside the parenthesis: -3 * (3/7)n = -9/7n and -3 * (-2/7) = 6/7.

The expression now becomes 1/7 - 9/7n + 6/7.

Combine like terms, which are the constant fractions: 1/7 + 6/7 = 7/7 = 1.

The final simplified expression is 1 - 9/7n.

Answer:

1 over 14

Step-by-step explanation:

The sectors labeled are B, Y, G, C, P, F, O, V, M, A, D, E, H, and R.

Total number of sectors = 14.

When Victoria spins the spinner 18 times, and it stops at the letter B 10 times.

When Victoria spins the spinner  the theoretical probability that this time it will stop at the letter B = Favourable  outcomes ÷total numner of outcomes.

Favourable outcomes =1 and Total number of outcomes = 14.

Hence Theoritical Probability = 1 over 14.

Answer:

r(p(x))=73

Step-by-step explanation:

To find the value of r(p(x)), the equation for p(x) which is 2x + 1, is substituted into every place that there is a x in the equation of r(x).

[tex]r(x)=4x-3\\r(p(x))=4(2x+1)-3\\r(p(x))=8x+4-3\\r(p(9))=8(9)+4-3\\r(p(9))=73[/tex]

I'm guessing its A:

If tomorrow is not my birthday, tomorrow is not Monday                        

Answer:

Step-by-step explanation:

Answer:

The answer would be A. He determined pounds per dollar by dividing 10 by 25 but wrote the unit rate as a dollar value.

Caitlin's reasoning is valid; in a triangle, the side opposite the largest angle is indeed the longest, according to the triangle inequality theorem.

Caitlin's reasoning that AC is the longest segment of fencing because it is opposite the largest angle measure in the figure is indeed valid. In any triangle, the longest side is always opposite the largest angle. This is due to the fact that the angles in a triangle are directly related to the lengths of their opposite sides.

The triangle side opposite the largest angle is the longest because of the triangle inequality theorem. So, if 68 degrees is the largest angle in the triangular piece of land that Caitlin is fencing, then segment AC, being opposite this angle, would be the longest side.

Caitlin's reasoning is valid based on the principle that in a triangle, the longest side is opposite the largest angle. The length of the third side can be determined using the cosine law or vector addition, depending on the given details.

The validity of Caitlin's reasoning is based on a principle in geometry that states the side opposite the largest angle in a triangle is the longest side. This fact is drawn from the Law of Sines which asserts that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Therefore, if AC is indeed opposite the largest angle, which is 68 degrees, then her reasoning is valid and AC would be the longest segment of fencing.

To calculate the length of the third side, one can use the cosine law or the vector addition method depending on the given information and the context of the question. If the internal angles and lengths of two sides are known, the cosine law allows for the calculation of the third side. In the context of vectors, the resultant vector, which in this case is the third side C, can be calculated by mathematically adding vectors A and B.

you were wrong, so I got it wrong...

The equation of the plane containing the two given lines is found by calculating the normal vector through the cross product of direction vectors and then using the normal vector with a point from the lines to form the plane's equation, which is 26x + y - 7z = 53.

To find the equation of a plane that contains two given lines, we must first find two direction vectors that represent the lines. In this case, we are given the lines r1(t) = (2, 1, 0) + t(1, 2, 4) and r2(t) = (2, 1, 0) + t(4, 1, 15). The direction vectors for these lines are (1, 2, 4) and (4, 1, 15), respectively. To find the normal vector to the plane, we take the cross product of these two direction vectors.

Let's calculate the cross product:
(1, 2, 4) * (4, 1, 15) = (2*15 - 4*1, 4*4 - 15*1, 1*1 - 2*4) = (26, 1, -7).

The normal vector to the plane is (26, 1, -7), which also represents the coefficients of the variables in the plane's equation. Since both lines pass through the point (2, 1, 0), we can use this point and the normal vector to find the plane's equation.

The general form of the plane's equation is 'Ax + By + Cz + D = 0', where (A, B, C) is the normal vector to the plane. Plugging in the point (2, 1, 0) and the normal vector (26, 1, -7), we get:

26*(x - 2) + 1*(y - 1) - 7*(z - 0) = 0
26x - 52 + y - 1 - 7z = 0
26x + y - 7z = 53

And this is the Equation to the required plane containing the two given lines.

Final answer:

The actual distance between Madison and Centerville is calculated by multiplying the map distance by the scale factor. Madison and Centerville are 66 km apart in real life.

Explanation:

The actual distance between Madison and Centerville is calculated by multiplying the map distance by the scale factor:

Map distance = 6 cm

Scale factor = 1 cm : 11 km

Actual distance = 6 cm * 11 km = 66 km

Therefore, Madison and Centerville are 66 km apart in real life.