Factor t12 – r3s15. (t4 – rs5)(t8 – rs5t4 – r2s10) (t4 – rs5)(t8 rs5t4 r2s10) (t9 – rs12)(t18 rs12t9 r2s24) (t4 rs5)(t8 – rs5t4 r2s10)

The general solution will be:

[tex]$$y(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right)+x^2-5 x-23$$[/tex]

[tex]$$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$$[/tex]

Consider the homogenous equation [tex]$14 y^{\prime \prime}+y^{\prime}+y=0$[/tex]

The auxiliary equation is [tex]$14 m^2+m+1=0$[/tex]

[tex]m=-\frac{1}{28}+i \frac{1}{28} \sqrt{55}[/tex]

The homogenous solution is

[tex]$$y_c(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right) \text {. }$$[/tex]

Consider the non-homogenous de [tex]$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$[/tex] by using the method of undetermined coefficients

consider [tex]$y_p=A x^2+B x+C \Rightarrow y_p^{\prime}=2 A x+B \Rightarrow y_p^{\prime \prime}=2 A$[/tex]

by plugging the values in [tex]$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$[/tex]

[tex]$$\begin{aligned}& 28 A+(2 A x+B)+A x^2+B x+C=x^2-3 x \\& \Rightarrow x^2(A)+x(2 A+B)+28 A+B+C=x^2-3 x\end{aligned}$$[/tex]

On comparing both sides we get

A=1.

2A+B =-3

=>B=-3-2A

=>B=-5

28A+B+C=0

=>C=-28A-B

=>C=-23

by plugging the values [tex]y_p=x^2-5 x-23[/tex].

The general solution will be:

[tex]$$y(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right)+x^2-5 x-23$$[/tex]

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The function rule for the functions f(x) and g(x) given by the table are;

f(x) = Log₁₀x

g(x) = 2ˣ

1) In the first table of the function f(x), we see the coordinates as;

(10, 1), (100, 2), (1000, 3), (10000, 4), (100000, 5)

Thus, we see that;

Log 10 = 1

Log 100 = Log 10² = 2 Log 10 = 2

Log 1000 = Log 10³ = 3Log 10 = 3

Thus, the function rule is Log₁₀x

2)  In the second table of the function g(x), we see the coordinates as;

(1, 2), (2, 4), (3, 8), (4, 16), (5, 32)

Thus, we see that;

2¹ = 2

2² = 4

2³ = 8

Thus, the function rule here is 2ˣ

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Answer:

Below

Step-by-step explanation:

x = a number

7 / (x+8)

One angle in a triangle is 10 degrees larger than the smallest angle, while the third angle is 10 degrees less than twice the smallest angle. As a result, the three angles are 55°, 45°, and 80°.

Three vertices and three sides make up a triangle. The angles of the triangle are formed by the connection of the three sides end to end at a single point. The sum of the triangle's three angles is 180 degrees.

Let's consider the three angles of the triangle as ∠P, ∠Q, and ∠R. Let's consider Q as the smallest angle. The sum of angles P, Q, and R is given as ∠P+∠Q+∠R=180°

Then, ∠P = ∠Q + 10° and ∠R = 2∠Q - 10°. Substituting these two angle values in the above equation,

[tex]\begin{aligned}\angle Q + 10^{\circ}+\angle Q+2\angle Q - 10^{\circ}&=180^{\circ}\\4\angle Q&=180^{\circ}\\\angle Q&=45^{\circ}\end{aligned}[/tex]

Then,

∠P = ∠Q + 10° = 45°+ 10° = 55°

∠R = 2∠Q - 10° = 2(45°) - 10° = 90° - 10° = 80°

The answers are 55°, 45°, and 80°.

The complete question is -

One angle of a triangle measure 10 degrees greater than the smallest angle and the third angle measure 10 degrees less than twice the smallest angle find the measure of the three angles.

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The lengths of the segments SH is 16, HM is 8, TH is 8, HR is 12,

TD = 12 and ER = 18.

A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.

We know a centroid divides a line segment in the ratio of 2 : 1 from vertices

to the other side.

Given, ΔSTR, H is the centroid EH = 6, DH = 4, and SM = 24.

SH = (2/3)×24 = 16.

HM = 8.

As DH = 4 hence TH = 8.

As EH = 6, HR = 12.

TD = (DH + TH) = 12.

ER = 6 + 12 = 18.

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a and C of the two-member frame=Ax=-300N,Ay=300N, cx=300n, cy=300N.

momentum  of all forces  about A is
=-600x1.5+Cxx3=0

cx=900/3=300n
cx=300n.

momentum of all forces about c is
-600x1.5-AXx3=0                  
Ax=-900/3=-300n
Ax=-300n

forces in Ab components is
600x1.5-Ayx3=0
ay=300n

momentum of all forces about b

600x1.5-ayx3-cyx3+cxx3=0
900-900-900+cxx3=0
cxx3=900
cx=300n

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The inequality that describes the problem is x < 180 × 4, which means that the total amount of money Mr. Dawson wants to spend on groceries in April should be less than $180 multiplied by 4, or $720.

Mr. Dawson is making a grocery budget for the month of April. He plans to split the budget equally among 4 shopping trips and wants to stay under budget. To figure out the maximum amount of money he should spend on groceries in April, he needs to calculate the total amount of money he should spend for all 4 shopping trips. To do this, he multiplies the maximum amount of money he should spend per trip, $180, by the number of trips, 4. This gives us $180 × 4 = $720. Therefore, Mr. Dawson should spend less than $720 on groceries in April. The inequality that describes the problem is x < 180 × 4, which means that the total amount of money Mr. Dawson wants to spend on groceries in April should be less than $720. This inequality ensures that Mr. Dawson will stay under budget and be able to complete his 4 shopping trips for the month of April.

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First, we need to find the total area of the land that is planted with cereals. To do this, we multiply the total area of the land, which is 2 hectares, by the percentage of the land that is planted with cereals, which is 100% - 20% - 10% = 70%. This gives us a total area of 2 hectares * 70% = 1.4 hectares.

Next, we need to divide the total area of 1.4 hectares into three parts, one for each type of cereal. Since 20% of the land is planted with corn, the area of land planted with corn is 1.4 hectares * 20% = 0.28 hectares. Similarly, since 10% of the land is planted with oats, the area of land planted with oats is 1.4 hectares * 10% = 0.14 hectares. Finally, the area of land planted with wheat is 1.4 hectares - 0.28 hectares - 0.14 hectares = 0.98 hectares.

Since 1 hectare is equal to 10,000 square meters, the area of land planted with corn is 0.28 hectares * 10,000 square meters/hectare = 2,800 square meters. The area of land planted with oats is 0.14 hectares * 10,000 square meters/hectare = 1,400 square meters, and the area of land planted with wheat is 0.98 hectares * 10,000 square meters/hectare = 9,800 square meters. So, each surface of land planted with corn is 2,800 square meters, each surface of land planted with oats is 1,400 square meters, and each surface of land planted with wheat is 9,800 square meters.

Answer:

See below.

Step-by-step explanation:

Given expression:

[tex]2(7^n) - 2(7^{n-1})+7^{n+1}[/tex]

[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\implies 2(7^n) - 2(7^{n-1})+7^{n} \cdot 7^1[/tex]

[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}[/tex]

[tex]\implies 2(7^n) - 2\left(\dfrac{7^{n}}{7^{1}}\right)+7^{n}\cdot 7^1[/tex]

Simplify:

[tex]\implies 2(7^n) - \dfrac{2}{7} (7^n)+7(7^{n})[/tex]

Factor out the common term 7ⁿ:

[tex]\implies \left(2- \dfrac{2}{7} +7\right)7^{n}[/tex]

Therefore:

[tex]\dfrac{61}{7}(7^n)[/tex]

A prime number is a whole number greater than 1 that cannot be made by multiplying other whole numbers.  Therefore, the factors of a prime number are 1 and the number itself.

If n is an integer greater than 1, the number will always have at least 4 factors (1, 7, 61 and itself) and therefore cannot be a prime number by definition.

For example:

[tex]n = 2 \implies \dfrac{61}{7}(7^2)=61 \times 7=427[/tex]

[tex]n = 3\implies \dfrac{61}{7}(7^3)=61 \times 7^2=2989[/tex]

Therefore, the largest prime number is when n = 1:

[tex]n = 1 \implies \dfrac{61}{7}(7^1)=61[/tex]

The cost of the drink after discount is 3.56 dollars.

You don't have enough money to buy the drink.

You are short of 0.89 dollars.

The price of a venti vanilla bean Frappuccino from Starbucks is $4.45, but you have only $3 with you. Your friend is working and gets you a discount of 20%.

The cost of the drink with discount can be calculated as follows:

amount discounted = 20% of 4.45

amount discounted = 20 / 100 × 4.45

amount discounted = 89 / 100

amount discounted =  0.89 dollars

Therefore,

cost of the drink after discount = 4.45 - 0.89

cost of the drink after discount = 3.56 dollars

Therefore, you don not have enough money to buy the drink.

You are short of 4.45 - 3.56 = 0.89 dollars.

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Answer:

Let the width of the rectangle be w.

Then the length of the rectangle is w+2.

The perimeter of the rectangle is the sum of all its sides, so it is 2(w) + 2(w+2) = 60.

Expanding the parentheses and combining like terms, we get:

2w + 2w + 4 = 60

4w + 4 = 60

4w = 56

w = 14

The width of the rectangle is 14.

The length of the rectangle is 14+2 = 16.

Therefore, the length of the rectangle is 16 and the width is 14.

The double integral of (2x)/(1+xy)dA over the rectangular region R = [0, 2]×[0, 1] give an approximate value of 2.0044.

Therefore, the answer is 2.0044.

By double Riemann sum,

∫∫(2x)/(1+xy) dA ≈ ∑(i = 1 to m)∑(j = 1 to n)f(xij*⁻, yij*⁻)ΔA

R = [0, 2]×[0, 1], that is a = 0, b = 2, c = 0, d = 1

Dividing R into four squares with m = n = 2 and corners of each square chosen (1, 1), (1, 2), ( 2, 1) and (2, 2). So midpoint of the square is (0.5, 0.5), (0.5, 1.5), ( 1.5, 0.5) and (1.5, 1.5)

Δx = (b - a)/m = 1

Δy = (d - c)/ n = 0.5

ΔA = ΔxΔy = 0.5

∫∫(2x)/(1+xy) dA ≈ ∑(i = 1 to 2)∑(j = 1 to 2)f(xij*⁻, yij*⁻)ΔA

= ΔA × (  f(0.5, 0.5) + f(0.5, 1.5) + f( 1.5, 0.5) + f(1.5, 1.5))

= 0.5 × ( (2×0.5)/(1+0.5×0.5) + (2×0.5)/(1+0.5×1.5) + (2×1.5)/(1+1.5×0.5) + (2×1.5)/(1+1.5×1.5))

≈ 0.5 × 4.0088

= 2.0044

--The question is incomplete, answering to the question below--

"Calculate the double integral.

(2x)/(1+xy) dA

R = [0, 2]×[0, 1]"

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Randy needs 8 loads.

To find the answer on the question we have to follow the method of multiplication and division .

A washing machine can hold [tex]\frac{3}{4}[/tex] kg of laundry in each load.

[tex]\frac{3}{4}[/tex] kg of laundry is equal to 1 load.

[tex]\frac{3}{4}[/tex] kg of laundry =  1 load

1 kg of laundry  =  [tex]\frac{1 * 4}{3}[/tex] load ( following the method of division )

1 kg of laundry = [tex]\frac{4}{3}[/tex] load

Randy has 6kg of laundry.

So, 1kg of laundry i.e [tex]\frac{4}{3}[/tex] load is to be multiplied by 6.

6kg of laundry = [tex]\frac{4 * 6}{3}[/tex] load ( following the method of multiplication)

                         = 8 loads

Hence, Randy needs 8 loads.

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All four measures of the angles are mentioned in the figure.

The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360 °.

If the measure of the angle is more than the right angle that is 90°. Then the angle is known as the obtuse angle.

Move the yellow dabs to make the portions cross. Allow the convergence to be called point V. Cause the lines to cross so that point ∠SVT is a harsh point under 135°.

Let the measure of the angle ∠SVU be 'x' which is greater than 45°. Then the measure of the angle ∠SVT will be '180 - x'.

All four measures of the angles are mentioned in the figure.

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Answer:

Sold:

adults: 375 tickets

student: 324 tickets

Step-by-step explanation:

a + s = 699       Eq. 1

a = s + 51          Eq. 2

a = adult tickets sold

s = student tickets sold

Replacing Eq. 2 im Eq. 1

(s+51) + s = 699

2s + 51 = 699

2s = 699 - 51

2s = 648

s = 648/2

s = 324

From Eq. 2

a = 324 + 51

a = 375

Check:

From Eq. 1:

a + s = 699

375 + 324 = 699

When a coin is weighted, there is a two-thirds chance that it will land on its head. suppose you toss this coin 15 times. let x represent the number of heads. Mean = 10 and Standard deviation = 2.88

The mean of x, which represents the number of heads when the coin is tossed 15 times, is 10. This is because the probability of getting heads is two-thirds, meaning that two out of every three tosses will result in heads.We apply the following formula to determine the standard deviation:

Standard Deviation = √(p*q*n), where p is the probability of getting heads, q is the probability of getting tails, and n is the number of tosses. In this case, p = 2/3, q = 1/3, and n = 15, so the standard deviation of x is 2.88.

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If d is directly proportional to t, then we can say that d = k * t, where k is a constant of proportionality. We can find the value of k by substituting the given values for d and t into this equation:

d = k * t

20 = k * 2

Solving for k, we get k = 10.

Now that we know the value of k, we can calculate the distance the car travels in 3 seconds by substituting 3 for t in the equation d = k * t:

d = 10 * 3

d = 30

Therefore, the car travels 30 meters in a time of 3 seconds.

Answer:

31

Step-by-step explanation:

7, 10, 12, 16, 19, 22, 25, 28, 31

Adding by 3

Hope this works

Answer:

-6p+12q

Step-by-step explanation:

−3(2p − 4q)

Distribute the -3

-3 * 2p -3 * -4q

-6p+12q

Answer:

[tex] \sf \: d) \: -6p + 12q [/tex]

Step-by-step explanation:

Now we have to,

→ simplify the given expression.

The expression is,

→ -3(2p - 4q)

Let's simplify the expression,

→ -3(2p - 4q)

→ -3(2p) - 3(-4q)

→ (-3 × 2)p - (3 × -4)q

→ -6p - (-12)q

→ -6p + 12q

Hence, answer is -6p + 12q.

Answer: When viewed from the side, the spire forms an isosceles triangle m, the area of the isosceles triangle is equal to  1/4b√4a²-b².

Step-by-step explanation:

What is isosceles triangle?

A triangle with two equal-length sides is said to be isosceles. The triangle's third side is referred to as the base, while the triangle's two equal sides are known as the legs. The triangle's legs are usually shorter than the base, which is usually the triangle's longest side.

To determine if the spire on top of a building forms an isosceles triangle when viewed from the side, we first need to define the three sides of the triangle. Let's call the two sides that are equal in length "s" and the third side "l".

Next, we need to determine if the triangle satisfies the definition of an isosceles triangle. The two sides "s" of the spire must therefore be of equal length for the spire to create an isosceles triangle.

To confirm this, we can measure the length of the sides "s" and compare them to each other. If they are equal in length, then the spire forms an isosceles triangle. If they are not equal in length, then the spire does not form an isosceles triangle.

For example, if the base of the triangle is 10 feet and the length of the two equal sides is 10 feet, then the triangle is isosceles. Therefore, the spire on the top of the building forms an isosceles triangle when viewed from the side.

Therefore, if the two sides "s" of the spire are equal in length, then the spire forms an isosceles triangle when viewed from the side.

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An isosceles triangle is a triangle with two sides of equal length and two equal angles.

What is isosceles triangle?

An isosceles triangle is a type of triangle that has two sides of equal length and two angles of equal measure. The two sides of equal length are the two legs of the triangle, while the third side is called the base. The angles opposite the two equal sides are equal in measure, while the angle opposite the base is the third angle. Isosceles triangles are classified according to the measure of their angles. An equilateral triangle is an isosceles triangle with all angles equal to 60°. An isosceles triangle with two angles equal to 45° is called an isosceles right triangle. An isosceles triangle with two angles equal to less than 45° is referred to as an acute isosceles triangle, while an isosceles triangle with two angles equal to more than 45° is referred to as an obtuse isosceles triangle. Isosceles triangles have many practical uses in engineering and architecture. For example, buildings and bridges often feature isosceles triangles in their roofs and arches in order to provide strength and stability.

When viewed from the side, the spire on the top of a building forms an isosceles triangle because it has two equal sides (the sides running up the spire) and two equal angles (the angles at the base of the spire).

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The correct answer is 4. Any x-values that result in the denominator being equal to 0.

When finding the domain of the quotient of two functions, we need to consider the domains of both the numerator and the denominator. The domain of a function is the set of all possible input values for which the function is defined. For the quotient of two functions, this means that the domain is the set of all x-values that are in the domains of both the numerator and denominator.

However, there is one additional restriction on the domain of the quotient of two functions. The denominator of the quotient cannot be equal to 0, because division by 0 is undefined. Therefore, any x-values that result in the denominator being equal to 0 are not included in the domain of the quotient.

In summary, the x-values that are not included in the domain of the quotient of two functions are any x-values that result in the denominator being equal to 0.

Approximately it will take 2 minutes to pass before they are 3 miles apart.

The time can be defined as the ratio of distance covered by an object to a unit speed. Time refers to the progression of events.

The Formula of Time of a given body can be expressed as,

Time = Distance÷Speed

one group takes a taxi uptown, traveling north at 35 miles per hour

The rest of the friends share another taxi, traveling south at 44 miles per hour

Suppose X as a time before they are 3 miles apart

so forming the equations:

35x+44x=3

79x=3

x=3/79hours.

To convert into minutes multiply x by 60 because 1 hour=60 min

x=3/79*60

x=0.0379*60

x=2.278

x=2 minutes.

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Answer:

-10

Step-by-step explanation:

f(x)=3x+2

f(-4)=3(-4)+2

f(-4)=-12+2

f(-4)=-10

Answer:

r = 6

Step-by-step explanation:

6г - 17r = -66

-(17г - 6r) = -66

-11r = -66

-11r/(-1) = -66/(-1)

11r = 66

11r/11 = 66/11

r = 66/11

r = 6

Answer:Isn't this cheating

Step-by-step explanation:

Answer:

f(-2) = -2

Step-by-step explanation:

Step 1: Substitute -2 for the variable n.

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

Step 2: Simplify the expression.

f(-2) = 4 - 6

Step 3: Evaluate the expression.

f(-2) = -2

Answer:

[tex]14 \frac{1}{8}[/tex]

Step-by-step explanation:

112 goes into 8 14 times, there is then 1/8 leftover

The match of vector field f for  f(x, y) = y, x is graph II .

We can say that some points for the given function are F(1,2)= (2,1) F(2,1) =(1,2) F(0,1) = (1,0) F(2,3) = (3,2) F(0,2)=(2,0) .

That means arrow heads should start with these points and this is shown accordingly in graph II only.

Therefore ,

option A matches with figure 2

option B matches with figure 1

option C matches with figure 3

option D matches with figure 4

A vector field is a way to visualize them as represented by a number of small arrows each having a magnitude or a direction.

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Answer:

y = 10 cm

Step-by-step explanation:

A = 12 · 5 / 2

A = 6 · 5 = 30 cm²

y = 30 · 2 / 6

y = 5 · 2

y = 10 cm