Select the correct answer from the drop-down menu. A vertical pole is supported by two ropes staked to the ground on opposite sides of the pole. One rope is 8 meters long, and the other is 7 meters. The distance between the stakes is 6 meters, and the height of the pole is meters.

Answer:

Our two numbers are:

[tex]2+4\sqrt{2} \text{ and } 4\sqrt{2}-2[/tex]

Or, approximately 7.66 and 3.66.

Step-by-step explanation:

Let the two numbers be a and b.

One positive real number is four less than another. So, we can write that:

[tex]b=a-4[/tex]

The sum of the squares of the two numbers is 72. Therefore:

[tex]a^2+b^2=72[/tex]

Substitute:

[tex]a^2+(a-4)^2=72[/tex]

Solve for a. Expand:

[tex]a^2+(a^2-8a+16)=72[/tex]

Simplify:

[tex]2a^2-8a+16=72[/tex]

Divide both sides by two:

[tex]a^2-4a+8=36[/tex]

Subtract 36 from both sides:

[tex]a^2-4a-28=0[/tex]

The equation isn't factorable. So, we can use the quadratic formula:

[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

In this case, a = 1, b = -4, and c = -28. Substitute:

[tex]\displaystyle x=\frac{-(-4)\pm\sqrt{(-4)^2-4(1)(-28)}}{2(1)}[/tex]

Evaluate:

[tex]\displaystyle x=\frac{4\pm\sqrt{128}}{2}=\frac{4\pm8\sqrt{2}}{2}=2\pm4\sqrt{2}[/tex]

So, our two solutions are:

[tex]\displaystyle x_1=2+4\sqrt{2}\approx 7.66\text{ or } x_2=2-4\sqrt{2}\approx-3.66[/tex]

Since the two numbers are positive, we can ignore the second solution.

So, our first number is:

[tex]a=2+4\sqrt{2}[/tex]

And since the second number is four less, our second number is:

[tex]b=(2+4\sqrt{2})-4=4\sqrt{2}-2\approx 3.66[/tex]

Answer:

[tex]2+4\sqrt{2}\text{ and }4\sqrt{2}-2[/tex]

Step-by-step explanation:

Let the large number be [tex]x[/tex]. We can represent the smaller number with [tex]x-4[/tex]. Since their squares add up to 72, we have the following equation:

[tex]x^2+(x-4)^2=72[/tex]

Expand [tex](x-4)^2[/tex] using the property [tex](a-b)^2=a^2-2ab+b^2[/tex]:

[tex]x^2+x^2-2(4)(x)+16=72[/tex]

Combine like terms:

[tex]2x^2-8x+16=72[/tex]

Subtract 72 from both sides:

[tex]2x^2-8x-56=0[/tex]

Use the quadratic formula to find solutions for [tex]x[/tex]:

[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex] for [tex]ax^2+bx+c[/tex]

In [tex]2x^2-8x-56[/tex], assign:

Solving, we get:

[tex]x=\frac{-(-8)\pm \sqrt{(-8)^2-4(2)(-56)}}{2(2)},\\x=\frac{8\pm 16\sqrt{2}}{4},\\\begin{cases}x=\frac{8+16\sqrt{2}}{4}, x=\boxed{2+4\sqrt{2}} \\x=\frac{8-16\sqrt{2}}{4}, x=\boxed{2-4\sqrt{2}}\end{cases}[/tex]

Since the question stipulates that [tex]x[/tex] is positive, we have [tex]x=\boxed{2+4\sqrt{2}}[/tex]. Therefore, the two numbers are [tex]2+4\sqrt{2}[/tex] and [tex]4\sqrt{2}-2[/tex].

Verify:

[tex](2+4\sqrt{2})^2+(4\sqrt{2}-2)^2=72\:\checkmark[/tex]

Answer:

y = 3x - 30

Step-by-step explanation:

First, put both equations into slope-intercept form.

x + 3y - 4 = 0

3y = -x + 4

y = (-1/3)x + (4/3)

2x + 5y - 20 = 0

5y = -2x + 20

y = (-2/5)x + 4

For a line to be perpendicular to y = (-1/3)x + (4/3), its slope should be the opposite and reciprocal of this line's slope.

The slope of y = (-1/3)x + (4/3) is (-1/3). The slope of a perpendicular line is 3.

To find the x-intercept of y = (-2/5)x + 4, plug in 0 for y and find x.

0 = (-2/5)x + 4

x = 10

The equation of the line right now is y = 3x + b

Plug in (10, 0) to find the y-intercept.

0 = 3(10) + b

b = -30

The equation of the line is y = 3x - 30

Answer:

12y

Step-by-step explanation:

Answer:

Option (1)

Step-by-step explanation:

We have to graph the system of inequalities given.

y > 2x + 1 -----(1)

y > |x| --------(2)

For inequality (1),

Solution area will be the area above the dotted line y = 2x + 1

Similarly, solution area of the second inequality will be the area above the dotted lines of y = |x|

Solution area of the system of inequalities will be the common area of both the graphs of the given system.

That will lie in quadrants 1 and 2.

Option (1) will be the answer.

Answer:

Associative property

Step-by-step explanation:

The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.

Hope this helps

Answer:

The solution to the system is (-2,6)

Step-by-step explanation:

I said the solution to the system is (-2,6) because that is where both lines cross, meaning that's the solution.

Hope this is hepful.

Answer:

4z²+4z-15 Square Metre  

Step-by-step explanation:

According To The Question,

Area Of Garden = Length × Breadth

Area Of Garden = (2z+5) × (2z-3)

Area Of Garden = 4z²-6z+10z-15

Area Of Garden=4z²+4z-15 Square Metre

Answer:

the answer is a 18 as 60 percent of 30 is 18

hope it helps

Answer:

(15+6)÷3

hope it helps!!

Given:

Principal value = $2300

Rate of interest = 14% compounded continuously.

To find:

The time taken by Brad's investment to triple.

Solution:

The formula for amount after the compound interest (Continuously) is:

[tex]A=Pe^{rt}[/tex]

Where, P is the principal, r is the rate of interest in decimal and t is the time period.

Triple of Brad's investment is

[tex]3\times \$ 2300=\$ 6900[/tex]

Substituting [tex]A=6900,P=2300\ r=0.14[/tex], we get

[tex]6900=2300e^{0.14t}[/tex]

[tex]\dfrac{6900}{2300}=e^{0.14t}[/tex]

[tex]3=e^{0.14t}[/tex]

Taking natural log on both sides, we get

[tex]\ln 3=\ln e^{0.14t}[/tex]

[tex]1.0986=0.14t[/tex]                [tex][\because \ln e^x=x][/tex]

[tex]\dfrac{1.0986}{0.14}=t[/tex]

[tex]7.84714=t[/tex]

After approximating the value, we get

[tex]t\approx 7.85[/tex]

Therefore, Brad's investment will take 7.85 years to triple.

Answer:

400 ft^2

Step-by-step explanation:

It can be shown that a square area is the most efficient way in which to use fencing.  If the area is not square, the area will inevitably be smaller.  

Calculus is the tool most often used in higher math to solve optimization problems.  

But the same goal can be achieved in this problem by working with constraints:  

If x and y are the length and width respectively, then

2x + 2y = 80 ft, or x + y = 40, or x = 40 - y.  This is one constraint.  

The other constraint involves the area:  A = x*y, or A = (40 - y)*y.  To maximize this, we need to rewrite (40 - y)*y in standard form:

A = 40y - y^2, or, finally, A = -y^2 + 40 y.  The coefficients of this quadratic are -1, 40 and 0; the axis of symmetry is thus

x = -b/ [2a], or, in this case, x = -40/[2*(-1)], or x = 20.

Thus, If x = 20, y = 20 also, proving that the shape of the enclosed yard is that of a square.

Then Mrs. L' 80 feet of fencing is sufficient to construct a 20 ft by 20 ft space, which comes out to a maximum area of 400 ft^2.

40 -

Answer:

Δy = 5  Δx = 1

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

Step-by-step explanation:

Answer:

[tex]-2 < 2x - 3 < 1 \ = \ \frac{1}{2} < x < 2[/tex]

Step-by-step explanation:

[tex]-2 < 2x - 3 < 1\\\\-2 + 3 < 2x - 3 + 3 < 1 + 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [ \ adding \ by\ 3 \ ]\\\\1 < 2x + 0 < 4\\\\1 < 2x < 4\\\\\frac{1}{2} < \frac{2x}{2} < \frac{4}{2} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [ \ divide \ by \ 2 \ ]\\\\\frac{1}{2} < x < 2[/tex]

Answer:

1/2 < x < 2

Step-by-step explanation:

-2<2x-3<1​

Add 3 to all sides

-2+3<2x-3+3<1​+3

1<2x<4

Divide all sides by 2

1/2 < 2x/2 <4/2

1/2 < x < 2

Answer:

t < -3

Step-by-step explanation:

-3t + 11 > 20

subtract 11 from both sides

-3t > 9

Divide both sides by -3

when multipying or dividing by a negative the inequality must be reversed

t < -3

Step-by-step explanation:

answer is in photo above

Answer:

-2< x< 8

Step-by-step explanation:

|x-3|<5 <=> -5<x-3<5 <=> -2<x<8

..

Answer:

-2x<8

Step-by-step explanation:

|x-3|<5<=> -5<x-3<5 <=>-2<x<8

.....

Answer: 251.2 inches.

Step-by-step explanation: You have to multiply 4*2*π*radius. So, simply multiply 4*2*3.14*10. It would come out as 251.2 inches.

Answer:

Check the text of the exercise, there is something wrong

Step-by-step explanation:

y=10x-4

y=-2x+10

These are two stright lines, they meet in one point only, P(7/6, 23/3)

Answer:

56 months 12+12x2=56 hope this helps

Step-by-step explanation:

Answer:

in this form, the "-r" would cause the result to decrease

I assume that the answer is "r"

if r = .1 (10 %) then 1-.1 = .9

if you have 100 items then y = 100(.9)^1 = 90

the total decreased by 10% ... y = 100(.9)^2 after 2 time periods

Step-by-step explanation:

Answer:

(x+y)+2

hope this helps

Answer:

you can do it

Step-by-step explanation:

by using your brain

Answer and Step-by-step explanation:

The answer is -7 (A.)

This is determined by looking at the graph. If you to to the x-point of -3 units, you will see there is a point at a y-value of -7 (units down).

#teamtrees #PAW (Plant And Water)

Answer:

the answer is c you will thank us later

Answer:

44

Step-by-step explanation:

divide 24 by 6 then multiply it by all the number then add it all

please make me brainliest

Answer:

5x=13.5

Step-by-step explanation:

Well, we know that 1x=2.7. So, if we want 5x we need to multiply 2.7*5 becuase 1x=2.7, so 5x=2.7*5.

Multiply. 2.7*5=13.5

So, 5x=13.5.

Hope this helps!

Answer:

13.5

Step-by-step explanation:

5x = ?

Replace 'x' with 2.7.

5(2.7) = ?

5(2.7) = 13.5

Hope this helps.

Answer:

1 square unit

Step-by-step explanation:

Area of the shaded region

= Area of the triangle with base 3 units and height 1 unit - Area of triangle with base 1 unit and height 1 unit.

= 1/2 *3*1 - 1/2 *1*1

= 1.5 - 0.5

= 1 square unit

Answer:

45kg

Step-by-step explanation:

60.3/4=45