Solve for a side in right triangles. Make sure to round your answer to the nearest hundredth

Answer:

Given equation:  [tex]y=x^2+1[/tex]

Therefore, we can say that any point on the curve has the coordinates [tex](a, a^2+1)[/tex]  (where a is any constant)

To find the gradient of the tangent to the curve at any given point, differentiate the equation.

Given equation:

[tex]y=x^2+1[/tex]

[tex]\implies \dfrac{dy}{dx}=2x[/tex]

Therefore, the gradient at point [tex](a, a^2+1)[/tex] is [tex]2a[/tex]

Using the point-slope form of linear equation, we can create a general equation of the tangent at point [tex](a, a^2+1)[/tex]:

[tex]\begin{aligned}y-y_1 & =m(x-x_1)\\ \implies y-(a^2+1)& =2a(x-a)\end{aligned}[/tex]

[tex]\implies y=2ax-2a^2+a^2+1[/tex]

[tex]\implies y=2ax-a^2+1[/tex]

Given that the tangents pass through point (-1, -3), input this into the general equation of the tangent:

[tex]\begin{aligned}y &=2ax-a^2+1\\ \implies -3 & =2a(-1)-a^2+1\end{aligned}[/tex]

[tex]\implies 0=-2a-a^2+1+3[/tex]

[tex]\implies a^2+2a-4=0[/tex]

Use the quadratic formula to solve for a:

[tex]\implies a=\dfrac{-2\pm\sqrt{2^2-4(1)(-4)}}{2(1)}[/tex]

[tex]\implies a=\dfrac{-2\pm2\sqrt{5}}{2}[/tex]

[tex]\implies a=-1 \pm \sqrt{5}[/tex]

Input the found values of a into the general equation of the tangent to create the equations of the two lines:

[tex]\begin{aligned}a=-1+\sqrt{5}\implies y & =2(-1+\sqrt{5})x-(-1+\sqrt{5})^2+1\\ y & =(-2+2\sqrt{5})x-(6-2\sqrt{5})+1\\ y & =(-2+2\sqrt{5})x+2\sqrt{5}-5 \end{aligned}[/tex]

[tex]\begin{aligned}a=-1-\sqrt{5}\implies y & =2(-1-\sqrt{5})x-(-1-\sqrt{5})^2+1\\ y & =(-2-2\sqrt{5})x-(6+2\sqrt{5})+1\\ y & =(-2-2\sqrt{5})x-2\sqrt{5}-5 \end{aligned}[/tex]

Therefore, the equations of the two lines that pass through the point (-1, -3) and are tangent to the graph of [tex]y=x^2+1[/tex] are:

[tex]y=(-2+2\sqrt{5})x+2\sqrt{5}-5[/tex]

[tex]y=(-2-2\sqrt{5})x-2\sqrt{5}-5[/tex]

Answer:

17. 125 ft^2

18. 9.4 in^2

Step-by-step explanation:

Both 17 and 18 are similar. You split that it into pieces.

Area of Rectangle/Square: Area = Length * Width

Area of a Triangle: Area = 1/2 * Base * Height.

17.

Area Rectangle:

Area = Length * Width

Area = 10 ft * 6 ft = 60 ft^2

Area Triangle:

Area = 1/2 * Base * Height

Area = 1/2 * 10 ft * 13 ft = 65 ft^2

Total Area = 60 ft^2 + 65 ft^2 = 125 ft^2

18.

Area Rectangle:

Area = Length * Width

Area = 2.7 in * 2 in = 5.4 in^2

Area Triangle:

Area = 1/2 * Base * Height

Area = 1/2 * 2 in * 4.4 ft = 4.4 ft^2

Total Area = 5.4 ft^2 + 4.4 ft^2 = 9.4 ft^2

radius= 7 ft

the circumference of the circle.

[tex]c = 2\pi \: r[/tex]

[tex]c = 2 \times \pi \times 7[/tex]

[tex]c = 43.9823[/tex]

[tex]c = 43.98 \: ft[/tex]

therefore, the circumference of the circle is 43.98 ft

Answer:

0.01296  to 5 decimal places.

Step-by-step explanation:

P(3) =  10! / 3! 7! * (0.68)^3 (1 - 0.68)^(10-3)

= 120 * (0.68)^3 * (0.32)^7

= 120 * 0.314432 * 0.00034359738

= 0.01296456

[tex]\sqrt[3]{4 a^4 b^5} = \sqrt[3]{4 \times 4^4 \times 2^5} = \sqrt[3]{(4\times 2)^5} = \sqrt[3]{(2^2\times2)^5} = \sqrt[3]{(2^3)^5} = \sqrt[3]{2^{15}} = \sqrt[3]{(2^5)^3} = 2^5 = 32[/tex]

[tex]\dfrac{a-c}{(b+c)^2} = \dfrac{4-(-5)}{(2+(-5))^2} = \dfrac{4+5}{(2-5)^2} = \dfrac9{(-3)^2} = \dfrac99 = 1[/tex]

Then the expression evaluates to 32 + 1 = 33.

The value of the expression

[tex]$\sqrt[3]{4a^4b^5} + \dfrac{a - c}{(b+c)^2}$[/tex]

is 33 when $a = 4$, $b = 2$, and $c = -5$.

Let's substitute the given values into the expression and calculate the result:

Given:

$a = 4$

$b = 2$

$c = -5$

Substituting the values:

[tex]$\sqrt[3]{4a^4b^5} + \dfrac{a - c}{(b+c)^2} = \sqrt[3]{4(4)^4(2)^5} + \dfrac{4 - (-5)}{(2+(-5))^2}$[/tex]

Calculating the values within the cube root:

[tex]$\sqrt[3]{4(4)^4(2)^5} = \sqrt[3]{4 \cdot 256 \cdot 32} = \sqrt[3]{32768} = 32$[/tex]

Calculating the values within the fraction:

[tex]$\dfrac{4 - (-5)}{(2+(-5))^2} = \dfrac{4 + 5}{(-3)^2} = \dfrac{9}{9} = 1$[/tex]

Now we can substitute the calculated values back into the expression:

[tex]$\sqrt[3]{4a^4b^5} + \dfrac{a - c}{(b+c)^2} = 32 + 1 = 33$[/tex]

Therefore, the value of

[tex]\sqrt[3]{4a^4b^5} + \frac{a - c}{(b+c)^2} \: is \: 33 \: when \: a = 4, \: b = 2, and \: c = -5[/tex]

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Answer:

24

Step-by-step explanation:

Use the triangle area formula: A=1/2 (base)(height)

Plug in what we know: 216=1/2(18)(h)

Solve for h: (1/2)(18)=9.

216/9=24.

So, h=24

To check, plug it into the formula: 1/2(18)(24)= 216

Answer:

8

Step-by-step explanation:

8+12+4+3+14+1+9+13=64, 64/8=8

Answer:

44.6

Step-by-step explanation:

First, to answer this question, we need to find the mean of this data set. When finding the mean, you use the same process you use when averaging.

So we would do: 8 + 12 + 4 + 3 + 14 + 1 + 9 + 13/8

Our new fraction turns into 64/8, which we can then divide to give us 52.625. We can round up our new number to give us 52.6

We now need to find the absolute value of the difference between each data value and mean. In simple terms, we use the answer we got from our fraction and subtract it from each number in our cluster of numbers/data set. Keep in mind when subtracting you will get some negative numbers, but in this scenario we take away any negative sings, so we don't end up with any negative numbers.

8 - 52.6 = -44.6 = 44.6

12 - 52.6 = -40.6 = 40.6

4 - 52.6 = -48.6 = 48.6

3 - 52.6 = -49.6 = 49.6

14 - 52.6 = -38.6 = 38.6

1 - 52.6 = -51.6 = 51.6

9 - 52.6 = -43.6 = 43.6

13 - 52.6 = -39.6 = 39.6

Finally, we need to to the same process we used in our first step, just use the new numbers we have instead of our old numbers.

This gives us 356.8/8, which when divided, will leave us with 44.6.

Therefore, our answer is 44.6

Answer:

x = 6

Step-by-step explanation:

Step 1. Rewrite in the form of y=mx+b

Step 2. Set y = 0 and solve for x

y - 6 = x - 12

y = x - 12 + 6

y = x - 6

0 = x - 6

6 = x

Answer:

y = x - 6

x = y + 6

Step-by-step explanation:

For y:

y - 6 = x - 12

y = x - 12 + 6

y = x - 6

For x:

y - 6 = x - 12

y - 6 + 12 = x

y + 6 = x

_______

Hope it helps ⚜

Answer:

164 ft

Step-by-step explanation:

The TIME of maximum height   (this is a dome shaped parabola)

        will be given by t = -b/2a = - 64 / (2)(-16) = 2 sec

Putting in t = 2   into the equation will give the max height

-16 (2^2) + 64(2) + 100 = max = 164 ft

Answer:

The answer is C and D.

Step-by-step explanation:

(-4)^1/2 = undefined

(-16)^1/4 = undefined

(-32)^1/5 = -2

(-8)^1/3 = -2

Answer:

117/108

Step-by-step explanation:

First, let's find the area of the shaded parts. Since the shaded squares and triangles are the same size, then all shaded squares have sides 3 in. by 3 in. because the shaded square in the middle has sides 3 in. by 3 in.

We can also see that the shaded triangles have legs 6 in. and 6 in. because one of the shaded triangles in the figure are labeled 6 in by 6 in.

Now we can find the area of the shaded square and triangle (area of a square is side^2 while the area of a triangle is base*height/2).

Shaded Square Area: 3^2 = 9 in^2

Shaded Triangle Area: 6*6/2 = 18 in^2

There are 5 shaded squares and 4 shaded triangles, so we can determine the shaded area now:

Shaded Area: 9*5 + 18*4 = 45 + 72 = 117 in^2

Now we need to find the area of the white rectangles and the area of the white triangles. We can see that the sides of the white rectangles are 6 in. and 3 in. We can also see that the sides of the white triangles are 3 in and 3 in.

Now we can find the area of the white rectangle and the white triangle.

White Triangle Area: 3*3/2 = 9/2 = 4.5 in^2

White Rectangle Area: 3*6 = 18 in^2

There are 4 white rectangles and 8 white triangles, so we can determine the white area now:

White Area: 4*18 + 8*4.5 = 72 + 36 = 108 in^2

The ratio of the area of the shaded pieces to the area of the white pieces is 117/108.

Answer:

It is larger by [tex]4*10^2[/tex]  times

Step-by-step explanation:

You use exponential division for this problem

first, divide 8 by 2

8/2 = 4

Then, look at 10^9 and 10^7

The bases of those numbers are the same, so you can just subtract the exponents since you're dividing.

[tex]10^9 / 10^7 = 10^{9-7} = 10^2[/tex]

Combine those two together to get:

4 * 10^2

Answer:

5mm

Step-by-step explanation:

Because, the radius is: It is the line between any point on the circle and the midpoint of the circle..

For this case we have the following expression:

[tex]5(y-6)[/tex]

We observe that the number 5 is a common factor of both terms within the parenthesis.

To prove it, we can apply the distributive property.

We have then:

[tex]5y-30[/tex]

We observe then that 5 is a common factor of the given expression.

Answer:

An algebraic expression that is a product with a factor of 5 is:

A) 5 (y-6)

Answer:

( 2, 60 )

Step-by-step explanation:

→ Read along the x - axis and then the y axis

Answer:

12/5 is the answer of your questions

thank you!!

Answer:

y=5

x=31

Step-by-step explanation:

5y=25

y=5

25+25+10x=360

50+10x=360

x=31

Answer:

$ 12 926.56

Step-by-step explanation:

If compounded annually:

  FV = 6000 ( 1 + .0525)^15 =

Answer:

Look up the website for those answers, I've gotten those type of worksheets before.

Step-by-step explanation:

There should be a pdf or document of the answers and if you can't find it I'll help you!

Answer

128

Step-by-step explanation:

72+56=128

You are welcome

Answer:

Step-by-step explanation:

h=b+2(assume h is hieght and b is base)

b*(b+2)=60

bb+2b=60

b=-1+√61

h=1+√61

Answer:

base=10

height=12

Step-by-step explanation:

Area of triangle formula is given by: A = (1/2)*b*h

where b: base of the triangle

           h: height of the triangle

Given that: h=b+2

                  A= 60 square yards

Solution:

60= (1/2)*b*(b+2)

[tex]\frac{b(b+2)}{2}[/tex] = 60

b²+2b=120

b²+2b-120=0

We got the quadratic equation: b²+2b-120=0

solve it to find b:  

Let x: coefficient of b² (x=1)

Let y: coefficient of b  (y=2)

Let z: constant (z= -120)

discriminant= y² - 4xz = 4 - 4(1)(-120) = 484

discriminant>0 so the equation has two roots:

b1= (-y-redical discriminant)/2x = (-2-redical(484))/2 = -12

b2= (-y+redical discriminant)/2x = (-2+redical(484))/2 = 10

b1= -12 is rejected because the base can't be negative

So b2=base=10

Now that we found the base, substitute to get the height:

h= b+2 = 10+2 =12

So height=12

Answer:

3rd option

Step-by-step explanation:

cos x = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{15}{17}[/tex]

They do through following ways

Answer:

Second and third one

Step-by-step explanation:

2 nd and third one ( are the same equation)

Answer:

bc it dose

Step-by-step explanation:

radius= 10 inches

3 revolutions in 2 seconds.

the linear velocity.

[tex]2\pi \: r = 2 \times \pi \times 10[/tex]

[tex] = 62.85inches[/tex]

[tex](3 \times 62.85)[/tex]

[tex] = 188.57[/tex]

[tex] \frac{188.87}{2} [/tex]

[tex] = 94.28[/tex]

hence, the linear velocity of the object is 94.28 in/s.

Answer:

$2.25 per lb

Step-by-step explanation:

x = 4.50/2

x = $2.25 per lb

The only solution that is true about the dependent system of equations is "There are infinitely many solutions".

Inconsistent System

For a  system of equations to have no real solution, the lines of the equations must be parallel to each other.

Consistent System

1. Dependent Consistent System

For a system of the equation to be a Dependent Consistent System the system must have multiple solutions for which the lines of the equation must be coinciding.

2. Independent Consistent System

For a system of equations to be an Independent Consistent System, the system must have one unique solution for which the lines of the equation must intersect at a particular.

As discussed above the Dependent consistent system has infinitely many solutions as their line are coinciding, therefore, the only solution that is true about the dependent system of equations is "There are infinitely many solutions".

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