Help please and fast

Answer:

The son is 8 years old

The father is 32 years old

Step-by-step explanation:

Let the man age be x

Let the son age = y

Right now the man is four times as older as his son = x=4y

In four years time he will be three times as old. ⇒x+4 =3(y+4)

Now substitute the value x=4y in x+4=3(y+4)

x+4=3(y+4)

4y+4=3(y+4)

4y+4=3y+12

Combine the like terms:

4y-3y =12-4

y=8

If the son is 8 years old than;

x=4y

x=4(8)

x=32

Father will be 32 years old....

Answer:

Present age of son: 8 years.

Present age of father: 32 years.

Step-by-step explanation:

Let x represent present age of the son and y represent present age of father.

We have been given that a man is four times as older as his son. 4 times of age of son would be [tex]4x[/tex].

We can represent this information in an equation as:

[tex]y=4x[/tex]

We are also told that in four years time he will be three times as old.

Age of father in 4 years would be [tex]y+4[/tex].

We can represent this information in an equation as:

[tex]y+4=3(x+4)[/tex]

Upon substituting [tex]y=4x[/tex] in 2nd equation, we will get:

[tex]4x+4=3(x+4)[/tex]

[tex]4x+4=3x+12[/tex]

[tex]4x+4-4=3x+12-4[/tex]

[tex]4x=3x+8[/tex]

[tex]4x-3x=3x-3x+8[/tex]

[tex]x=8[/tex]

Therefore, the present age of son is 8 years.

Upon substituting [tex]x=8[/tex] in equation [tex]y=4x[/tex], we will get:

[tex]y=4x\Rightarrow 4(8)=32[/tex]

Therefore, the present age of father is 32 years.

Answer:

[tex]\displaystyle 256[/tex]

Step-by-step explanation:

PEMDAS

P-parenthesis, E-exponent, M-multiply, D-divide, A-add, and S-subtracting.

Exponent rule: [tex]4^8^-^4[/tex]

Subtract by the exponent from left to right.

[tex]\displaystyle 8-4=4[/tex]

[tex]\displaystyle 4^4=4*4*4*4=256[/tex]

256 is the correct answer.

moving 4 to the other side x= 4 × -3 =-12

so x =-12

Answer:

x  ≤ -12

Step-by-step explanation:

1/4 x ≤ -3

Multiply by 4 on both sides to get rid of the fraction coefficient.

x  ≤ -12

This means x is x is equal to or less than -12

Solution:

The formula for tan(A+B) and tan(A-B) are:

[tex]tan(A+B) = \frac{tan(A)+tan(B)}{1-tan(A)tan(B)} \\\\tan(A-B) = \frac{tan(A)-tan(B)}{1+tan(A)tan(B)}[/tex]

The left hand side of the given expression is:

[tex]tan(\frac{\pi}{4}-\alpha ) tan(\frac{\pi}{4}+\alpha )[/tex]

Using the formula above and value of tan(π/4) = 1, we can expand this expression as:

[tex]tan(\frac{\pi}{4}-\alpha ) tan(\frac{\pi}{4}+\alpha )\\\\ = \frac{tan(\frac{\pi}{4} )-tan(\alpha)}{1+tan(\frac{\pi}{4} )tan(\alpha)} \times \frac{tan(\frac{\pi}{4} )+tan(\alpha)}{1-tan(\frac{\pi}{4} )tan(\alpha)}\\\\ = \frac{1-tan(\alpha)}{1+tan(\alpha)} \times \frac{1+tan(\alpha)}{1-tan(\alpha)}\\\\ = 1 \\\\ = R.H.S[/tex]

Thus, the left hand side is proved to be equal to right hand side.

Answer:

blue = 35

green = 29

red = 38

Step-by-step explanation:

Let r = red

Let g = green

let b = blue

r + g + b = 102

r = b + 3

b = g + 6    Subtract 6 from both sides of the equation

b - 6 = g

===================

substitute for red and green

(b + 3) + b + b - 6 = 102

b + 3 + b + b - 6 = 102

Combine like terms

3b - 3 = 102

Add 3 to both sides

3b - 3 + 3 = 102 + 3

3b = 105

Divide by 3

3b/3 = 105/3

b = 35

======================

r = b + 3

r = 35 + 3

r = 38

=======================

g = b - 6

g = 35 - 6

g = 29

========================

Answer:

red=35  green=29     blue=38

Step-by-step explanation:

[tex]\bf \begin{cases} f(x)=&5x-1\\ f(-3)=&5(-3)-1\\ f(-3)=&-15-1\\ f(-3)=&-16 \end{cases}\qquad \begin{cases} g(x)=&2x^2+1\\ g(-3)=&2(-3)^2+1\\ g(-3)=&2(-3)(-3)+1\\ g(-3)=&2(9)+1\\ g(-3)=&18+1\\ g(-3)=&19 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ (f\times g)(-3)\implies f(-3)\times g(-3)\implies (-16)(19)\implies -304[/tex]

Answer:

a) 132.0 mmHg, b) 50 years old.

Step-by-step explanation:

a) Plug in 48 where you see the letter y and simplify, preferably with a calculator.

P = 0.005(48)^2 - 0.01(48) + 121

P = 132.04 mmHg, to the nearest tenth would be 132.0 mmHg

b) Plug in 133 for P and solve for y.

0.005y^2 - 0.01 + 121 = 133

To make it a little easier on myself -- and because I haven't practiced a diff. method in a while -- I simplified the equation to 0.005y^2 - 0.01y - 12 = 0 by subtracting 133 from both sides. I did that so that I can could then use the quadratic formula to solve.

Quadratic formula is y = (-b +/- √(b^2 - 4ac)) / 2

Now we plug in our given information, that new trinomial, to solve for y

[tex]y = \frac{0.01 +/- \sqrt{(0.01)^2 - 4(0.005)(-12)} }{2(.0.005)} \\y = \frac{0.01 +/- \sqrt{0.2401}}{0.01}[/tex]

[tex]y = \frac{0.01}{0.01} +/- \frac{\sqrt{0.2401}}{0.01} \\y = 1 +/- \frac{0.49}{0.01}\\y = 1+/- 49[/tex]

Because it is a trinomial, you are given two answers. You get y = 48 and y = 50. In order to find out which is right, you plug in and see which on yields 133 as the answer. Given the part a), I already know it's not 48. When I plug in 50, I get 133. Therefore, 50 years old is your answer.

The systolic pressure for a person who is 48 years old is approximately 132.0 mm Hg. If a person's systolic pressure is 133 mm Hg, their age is approximately 49 years when rounded to the nearest whole year.

A) Systolic Pressure Calculation for Age 48

To find the systolic pressure for a person who is 48 years old, we use the given equation:

P = 0.005y^2 - 0.01y + 121

Substitute y = 48 into the equation:

P = 0.005(48)^2 - 0.01(48) + 121 = 0.005(2304) - 0.48 + 121 = 11.52 - 0.48 + 121 = 11.04 + 121 = 132.04 mm Hg

To the nearest tenth, the systolic pressure is 132.0 mm Hg.

B) Age Calculation for Systolic Pressure of 133 mm Hg

To find the age when a person's systolic pressure is 133 mm Hg, we set P = 133 and solve for y:

0.005y^2 - 0.01y + 121 = 133 0.005y^2 - 0.01y - 12 = 0

Using the quadratic formula, y = [-b ± √(b^2 - 4ac)] / (2a), where:

a = 0.005,
b = -0.01, and
c = -12.
We find the positive root that makes physical sense for age:

y ≈ 48.9 years

The age rounded to the nearest whole year is 49 years.

The numbers ordered from least to greatest are -0.52, -5/16, -1/5, 3/4, and 0.90. Negative values are ordered by ascending absolute value, while positive values are ordered normally.

To order the numbers from least to greatest, we first need to compare the negative numbers, then the positive fractions and decimals. It's important to understand that negative numbers are less than zero, and the number with the largest absolute value is actually the smallest when negative. Positive numbers are greater than zero, with larger decimal or fractional values representing larger numbers.

The correct order from least to greatest is: -0.52, -5/16, -1/5, 3/4, and 0.90.

Answer:

Step-by-step explanation:

Square base of the base edge = 12 meters and slant height = 6√2 meters

Apothem of the right pyramid will be = AC

1). Now AC = [tex]\sqrt{AB^{2}-BC^{2}}[/tex]

AC = [tex]\sqrt{(6\sqrt{2})^{2}-(6)^{2}}[/tex]

     = [tex]\sqrt{72-36}=\sqrt{36}[/tex]

     = 6

Now Apothem = 6 meters

2). Hypotenuse of Δ ABC = AB = 6√2 meters

3). Height AC = 6 meters

4). Volume of the pyramid = [tex]\frac{1}{3}(\text{Area of the base})(\text{Apothem})[/tex]

Volume = [tex]\frac{1}{3}(12)^{2}(6)[/tex]

             = 2×144

             = 288 meter²

Correct responses:

The given dimensions of the right pyramid having a square base are;

Base edge length, l = 12 meters

Slant height = 6·√3

Required:

Length of the apothem.

Solution:

The apothem, a, is the line drawn from the middle of the polygon to the midpoint of a side.

Therefore;

The apothem of the square base = [tex]\dfrac{l}{2} [/tex] = [tex]\overline{BC}[/tex]

Which gives;

[tex]a = \dfrac{12}{2} = 6[/tex]

Required:

The hypotenuse of triangle ΔABC

Solution:

The hypotenuse of ΔABC = The slant height of the square pyramid = 6·√2 meters

Therefore;

Required:

The height of the pyramid

Solution:

The height of the pyramid = The length of the side [tex]\overline{AC}[/tex] in right triangle

ΔABC, therefore, by Pythagorean theorem, we have;

[tex]\overline{AC}^2[/tex] = [tex]\overline{AB}^2[/tex] - [tex]\overline{BC}^2[/tex]

Which gives;

[tex]\overline{AC}^2[/tex] = (6·√2)² - 6² = 6²·((√2)² - 1) = 6² × 1  = 6²

[tex]\overline{AC}[/tex] = √(6²) = 6

Required:

The volume of the pyramid.

Solution:

[tex]The \ volume \ of \ a \ pyramid \ is, \ V = \mathbf{\dfrac{1}{3} \times Base \ area \times Height}[/tex]

[tex]V = \dfrac{1}{3} \times A \times h [/tex]

The base area of the square pyramid, A = 12 m × 12 m = 144 m²

Therefore;

[tex]V = \dfrac{1}{3} \times 144\, m^2 \times 6 \, m = 288 \, m^2[/tex]

Learn more about the volume of solids here:

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Answer:

There are 4 boys and 17 girls in the class.

Step-by-step explanation:

Out of 21 students in Mrs. Clark's class, 5th of the students are boys and rest are girls.

out of 21 every 5th student is boy, so total boys are = [tex]\frac{21}{5}[/tex]

                                                                                      = [tex]4\frac{1}{5}[/tex]

The whole number is 4, so the number of boys are 4.

The number of girls = 21 - 4 = 17 girls

There are 4 boys and 17 girls in the class.

Answer:

Here is a complete list of numbers that 6501 is divisible by:

1, 3, 11, 33, 197, 591, 2167, 6501

Step-by-step explanation:

I hope that was the answer you were looking for have a nice day :p

The number 6501 is not divisible by common divisors like 2, 3, 4, etc., and is in fact a prime number.

To determine the divisibility of the number 6501, we examine the number against known divisibility rules. However, unlike numbers like 4, 12, and 11, which have easy-to-apply divisibility rules, 6501 does not have an obvious rule that we can apply. So, we have to actually perform the division or use a calculator if we are trying to determine its divisibility by numbers other than 1 and itself.

For example:

The number 6501 is divisible by 1 and 6501 (since all numbers are divisible by themselves and 1).

To check for divisibility by other numbers, use division (for example: 6501 ÷ 2 is not an integer, so it's not divisible by 2).

We can conclude that 6501 is a prime number because other than 1 and 6501 itself, there are no other numbers that divide it evenly, indicating no divisors that give a whole number as the result.

Answer:

32

Step-by-step explanation:

I assume you want to find the pills per month?  You know that you take 8 pills per week, and there are 4 weeks in a month.  So:

8 pills/week × (4 week / month) = 32 pills/month

Check the picture below.

The greatest distance between the balls placed by Alice on a straight line is between the balls at points P and R, which equates to 18 units.

The point coordinates given for the balls that Alice placed are located on a straight line. They are vertically aligned, as the x-coordinate is the same (-5) for all the balls. The y-coordinate represents the vertical position of each ball.

To find the greatest distance between the balls, calculate the distance between the highest point P (-5,9) and the lowest point R (-5,-9). The formula to calculate the distance between two points (x1, y1) and (x2, y2) on a straight line is sqrt[(x2 - x1)^2 + (y2 - y1)^2].

However, as x1 and x2 are the same, the calculation simplifies to |y2 - y1|. So the distance between P and R would be |-9 - 9| which equals 18 units. Therefore, the two balls at points P and R are separated by the greatest distance.

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[tex]\bf \begin{array}{ccll} gallons&minutes\\ \cline{1-2} 4.39&1\\ 280&x \end{array}\implies \cfrac{4.39}{280}=\cfrac{1}{x}\implies 4.39x=280\implies x=\cfrac{280}{4.39} \\\\\\ x\approx \stackrel{\textit{minutes}}{63.78}~\hspace{7em}63.78 ~~\begin{matrix} min \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ \cdot \cfrac{hr}{60 ~~\begin{matrix} min \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ } \implies \boxed{1.063~hr}[/tex]

Answer:

328 pages for the novel

123 pages for the second day

Step-by-step explanation:

Let the number of pages of the novel = x

Raw Equation

(1/4)x + 1/2 (3/4)x + 123 = x

Solution

(1/4)x + (3/8)x+ 123 = x

Change the fractions to common denominators.

(2/8)x + (3/8)x + 123 = x

Add the fractions.

(2/8 + 3/8)x + 123 = x

Subtract (5/8)x from both sides.

(5/8)x + 123 = x

(5/8)x- (5/8)x + 123 = x - (5/8)x

Multiply both sides by 8

123 = (3/8)x  

123 * 8 = 3x

Divide by 3

984 = 3x

984/3 = 3x / 3

328 = x

===================

At the end of the second day, she read 3/8 * 328 = 123 pages.

The novel has 328 pages. By the end of the second day, the student had read 205 pages of the novel.

The student's schoolwork question can be addressed by setting up and solving algebraic equations. Let's denote the total number of pages in the novel as x. On the first day, one-quarter of the novel is read, which is x/4 pages. So, there are 3x/4 pages remaining. On the second day, half of the remaining pages are read, which is (1/2) × (3x/4) = 3x/8 pages. On the third day, the student reads the last 123 pages, which were all the pages that were left. Therefore, the equation to solve for x is:

x - (x/4 + 3x/8) = 123

We can solve this equation to find out the total number of pages in the novel:

First, let's find a common denominator for the fractions. It is 8.

8x/8 - (2x/8 + 3x/8) = 123

8x/8 - 5x/8 = 123

3x/8 = 123

Let's multiply both sides of the equation by 8/3 to solve for x.

x = 123 × (8/3)

x = 328

The novel has 328 pages.

To find out how many pages were read by the end of the second day, we add the amount read on the first and second days:

(x/4) + (3x/8) = (2x/8) + (3x/8) = 5x/8

5x/8 when x = 328 is:

(5 × 328)/8 = 205

By the end of the second day, 205 pages were read.

Answer:

Step-by-step explanation:

The formula of a volume of a pyramid:

[tex]V=\dfrac{1}{3}BH[/tex]

B - base area

H - height

In the base we have a square withe side s = 2cm

The formula of an area of a square with side s:

[tex]A=s^2[/tex]

Substitute:

[tex]A=2^2=4\ cm^2[/tex]

The height H = 3.75 cm.

Calculate the volume:

[tex]V=\dfrac{1}{3}(4)(3.75)=\dfrac{15}{3}=5\ cm^3[/tex]

Answer: C. 58 1/3 cm∧2

Just took quiz.

Answer:

least possible number of sweets = lowest common multiple of 5,6 & 10 - 2

-I hope this helps! I got it figured out until near like the very end.-

-Please mark as brainliest!- Thanks!

Final answer:

The smallest number of sweets that satisfies the condition of being left with certain remainders when shared with different numbers of friends is found using modular equations, with the solution being 59 sweets.

Explanation:

To solve the problem presented for Vivian and her sweets, we will use the concept of simultaneous congruences from number theory.

Vivian's sweets when divided by 4 leave a remainder of 3, when divided by 5 leave a remainder of 4, and when divided by 9 leave a remainder of 8. This situation translates to the following set of modular equations:

The smallest number that satisfies all these conditions is known as the least common multiple plus the respective remainders.

With the aid of the Chinese Remainder Theorem, we can conclude that the smallest number of sweets that satisfies all conditions is 59. This is the least number of sweets Vivian can have and still meet the conditions given for sharing among her friends.

Answer:

Answer is C

Step-by-step explanation:

The root is placed in the denominator section, and the exponent is placed in the numerator section. it is a fraction because you are not using the entire exponent, you will only be using part of it as the other part of the exponent is negated by the X

For this case we have that by properties of roots and powers it is fulfilled that:

[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]

So, if we have the following expression:

[tex]\sqrt [3] {8 ^ x}[/tex]

According to the given definition, we can rewrite it as:

[tex]8 ^ {\frac {x} {3}}[/tex]

ANswer:

[tex]8 ^ {\frac {x} {3}}[/tex]

Option C

Answer:

3√3 = l

√3 = w

Step-by-step explanation:

l = 3w

9 = 3w[w] ↷

9 = 3w² [Divide by 3]

3 = w² [Take the square root]

√3 = w [plug this back into the top equation to get a length of 3√3]

[3√3][√3] = 9 [Area]

I am joyous to assist you anytime.

Answer:

3 + 2i

Step-by-step explanation:

3 + √(-4) = 3 + i√4 = 3 + 2i

Recall that √(-1) = i

Answer:

The catcher will have to throw roughly 127 feet from home to second base.

Step-by-step explanation:

We have to use the Pythagorean Theorem. This gives us the third side length if we know the other two side lengths.

a² + b² = c²

90² + 90² = c²

8100 + 8100 = c²

16200 = c²

[tex]\sqrt{16200}[/tex] = [tex]\sqrt{c^2}[/tex]

c = 127.28 ft

Answer:

B. (x-4)^2 + (y - 3)^2 = 5^2

Step-by-step explanation:

The equation for a circle is given by

(x-h)^2 + (y-k) ^2 = r^2

where (h,k) is the center and r is the radius

We have a center of (4,3) and a radius of 5

(x-4)^2 + (y-3) ^2 = 5^2

Answer:

p=8

Step-by-step explanation:

We know that 'p' represents a digit.

We can say that:

x + 5/17 = 1

Solving for 'x' we have:

x = 1 -5/17

x = 12/17

Now, multiplying 'x' by 4, we have:

x = 48/68

Therefore, p=8

Answer:

30 square units

Step-by-step explananation:

First of all we need to know the formula to finding the perimeter of a rectangle which is:

P = 2 x L(Length) + 2 x h(Height)

12 and 3 are apart by 12 - 3

12 - 3 = 9

Then, we subtract 9 from 3 ( :

9 - 3 = 6)

To get 6 as our answer.

6 will be the width and 9 will be the length.

Now we solve for the perimeter by plugging in our values into our formula:

P = 2(9) + 2(6)

P = 18 + 12

P = 30 square units

Answer:

x= -2/5 and y=13/5

Step-by-step explanation:

Lets assume that the larger number = x

And the smaller number = y

According to the given statement three times a number added twice a smaller number is 4, it means;

3x+2y=4 -------- equation 1

Now further twice the smaller number less than twice the larger number is 6,it means;

2y-2x=6 --------equation 2

Solve the equation 2.

2y=2x+6

y=2x+6/2

y=2(x+3)/2

y=x+3

Substitute the value of y=x+3 in the first equation.

3x+2y=4

3x+2(x+3)=4

3x+2x+6=4

Combine the like terms:

5x=4-6

5x=-2

x= -2/5

Put the value x= -2/5 in equation 2.

2y-2x=6

2y-2(-2/5)=6

2y+4/5=6

By taking L.C.M we get

10y+4/5=6

10y+4=6*5

10y+4=30

10y=30-4

10y=26

y=26/10

y=13/5

Hence x= -2/5 and y=13/5....

To find the numbers, we first solve for x and y using a system of equations. We find that x is 2 and y is -1. Thus, the larger number is 2 and the smaller number is -1.

Detailed Explanation is as follows:

Let the larger number be x and the smaller number be y. Based on the problem, we can set up the following system of equations:

3x + 2y = 4

2x - 2y = 6

Add the equations together to eliminate y.

3x + 2y + 2x - 2y = 4 + 6

5x = 10

x = 2

Now, Substitute x back into one of the original equations

Using the first equation:

3(2) + 2y = 4

6 + 2y = 4

2y = -2

y = -1

Therefore, the larger number is 2 and the smaller number is -1.

Hence the larger number is 2 and the smaller number is -1.

[tex]\displaystyle\\\text{Decompose numbers into prime factors.}\\\\32=2^5\\\\48=2^4\times3\\\\64=2^6\\\\\text{greatest common divisor (gcd)~}~=2^4=\boxed{\bf16 m}\\\\\boxed{\text{\bf The maximum possible length of each wire cut is 16 m.}}[/tex]

The maximum possible length for each cut of wire that Mrs. Agustin can make from the 32m, 48m, and 64m wire coils is 16m.

In this problem, Mrs. Agustin is aiming to cut three different lengths of wires into equal parts. Therefore, the maximum possible length of each cut of wire can be calculated by finding the greatest common divisor (GCD) of the lengths of the three wires. The lengths of the wire coils are 32 m, 48 m, and 64 m. The GCD of these numbers is 16 m.

Thus, the maximum possible length of each cut wire would be 16 m.

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Answer:

[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{12})+\cos(\frac{5\pi}{12}))[/tex]

So the blank is cos.

Step-by-step explanation:

There is an identity for this:

[tex]\cos(a)\cos(b)=\frac{1}{2}(\cos(a+b)+\cos(a-b))[/tex]

Let's see if this is fit by your left hand and right hand side:

So [tex]a=\frac{\pi}{4}[/tex] while [tex]b=\frac{pi}{6}[/tex].

Let's plug these in to the identity above:

[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{4}+\frac{\pi}{6})+\cos(\frac{\pi}{4}-\frac{\pi}{6}))[/tex]

Ok, we definitely have the left hand sides are the same.

Let's see if the right hand sides are the same.

Before we move on let's see if we can find the sum and difference of [tex]\frac{\pi}{4}[/tex] and [tex]\frac{\pi}{6}[/tex].

We will need a common denominator.  How about 12? 12 works because 4 and 6 go into 12.  That is 4(3)=12 and 6(2)=12.

[tex]\frac{\pi}{4}+\frac{\pi}{6}=\frac{3\pi}{12}+\frac{2\pi}{12}=\frac{5\pi}{12}[/tex].

[tex]\frac{\pi}{4}-\frac{\pi}{6}=\frac{3\pi}{12}-\frac{2\pi}{12}=\frac{\pi}{12}[/tex].

Let's go back to our identity now:

[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{4}+\frac{\pi}{6})+\cos(\frac{\pi}{4}-\frac{\pi}{6}))[/tex]

[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{5\pi}{12})+\cos(\frac{\pi}{12}))[/tex]

We can rearrange the right hand side inside the ( ) using commutative property of addition:

[tex]\cos(\frac{\pi}{4})\cos(\frac{\pi}{6})=\frac{1}{2}(\cos(\frac{\pi}{12})+\cos(\frac{5\pi}{12}))[/tex]

So comparing my left hand side to their left hand side we see that the blank should be cos.

This is a trigonometric equation requiring the use of sum-to-product identities. By applying the identity and simplifying, the result is: cos(pi/4)cos(pi/6) = 1/2(cos pi/12 + cos 5pi/12). The blank should be filled by a '+' sign.

The question refers to a trigonometric identity equation in the field of mathematics. To solve it, we have to use the principle of the sum-to-product identities from trigonometry.

Let's use the identity: cos(A)cos(B) = 1/2 [cos(A - B) + cos(A + B)]. In this case, A = pi/4 and B = pi/6.

Therefore, cos(pi/4)cos(pi/6) = 1/2 [cos(pi/4 - pi/6) + cos(pi/4 + pi/6)].

Simplified further cos(pi/24) + cos(5pi/12).

So, the blank should be filled with a '+', making the equation cos(pi/4)cos(pi/6) = 1/2(cos pi/12 + cos 5pi/12).

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Answer:

9 m

Step-by-step explanation:

If x is the length of the original square, then x+5 is the length of the new square.

Area of a square is the square of the side length, A = s².  Since the area of the new square is 196 m²:

196 = (x + 5)²

Solving, first take the square root of both sides:

±14 = x + 5

Subtract 5 from both sides:

x = -5 ± 14

x = -19, x = 9

x can't be negative, so x = 9.  The side length of the original backyard is 9 m.

Answer:

9m

Step-by-step explanation:

If Mrs. G is planning an expansion of her square back yard and side of the original backyard is increased by 5 m, the new total area of the backyard will be 196 m2. The length of each side of the original backyard is 9m.

x = length of original square

x + 5

Formula: A = s²

Therefore, the side length of the original backyard is 9 m.

Answer:

A. The center of dilation is point C.

B.  It is a reduction.

E.  The scale factor is 2/5.

Step-by-step explanation:

The center is shown as C.  You can see this from the line segments they drew through C, the image, and the pre-image.

The pre-image is the image before the dialation.  The pre-image here is XYZ.

The image is the image after dialation. The image is X'Y'Z'.

If you look at the pre-image XYZ and then it's image X'Y'Z', ask yourself the image get smaller or larger.  To me I see a larger triangle being reduce to a smaller triangle so this is a reduction.

The scale factor cannot be bigger than 1 because the image shrunk so D is definitely not a possibility.

E. is a possibility but let's actually find the scale factor to see.

We can calculate [tex]\frac{CX'}{CX}[/tex] or [tex]\frac{CY'}{CY}[/tex] or [tex]\frac{CZ'}{CZ}[/tex] to find out what the scale factor is.

[tex]\frac{CX'}{CX}=\frac{2}{5}[/tex]

[tex]\frac{CZ'}{CZ}=\frac{3}{7.5}=\frac{2}{5}[/tex].

The scale factor is 2/5.

Answer:

A, The center of dilation is point C.

B, It is a reduction.

and E, The scale factor is 2/5.

Step-by-step explanation:

Answer:

1 hour.

Step-by-step explanation:

You have to do the total distance /total speed together.

You have to do this to ensure that you are counting how fast they are travelling together.

(450 km) / the sum of their speed (240+210)

450/ (240+210)  

=450/ 450  

=1

Answer: 1 Hour.

Answer:

t = 1 hour

Step-by-step explanation:

Oddly the distances add. Each plane will contribute a certain distance to make the 450 km. They will meet after the same number of hours have passed.

d = r * t

r1 = 240 km/hour

t1 = t

r2 = 210 km/hour

t2 = t

240*t + 210*t = 450     collect the terms on the left.

450t = 450                   divide by 450

450t/450 = 450/450

t =1

After 1 hour they will meet.