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marusya05 [52]
3 years ago
9

Let p and q be rational. Prove that p+q is rational

Mathematics
1 answer:
Contact [7]3 years ago
4 0

Let p and q rational numbers (what does it means?), then, p and q coul be written as

p = \frac{a}{b},a, b \in \mathbb{Z}, b \neq 0\\\\q = \frac{c}{d}, c,d \in \mathbb{Z}, d \neq 0

then,

p +q = \frac{a}{b} + \frac{c}{d} = \frac{ad +bc}{bd}

and we know that ad+bc is an integer, so, p +q is a rational number.

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A lacrosse player throws a ball into the air from a height of 6 feet with an initial vertical velocity of 64 feet per second. Wh
-Dominant- [34]

Answer:

Step-by-step explanation:

I'm going to use calculus to solve this, because it's the simplest way.  

The acceleration due to gravity in feet is the second derivative of the position function.  We will start with the acceleration and work backwards with antiderivatives to get to the position function.

a(t) = -32.  Going backwards and using the fact that the initial vertical velocity is 64 ft/sec, our velocity function is

v(t) = -32t + 64.  Going backwards and using the fact that the initial height of the ball is 6 feet, our position function is

s(t)=-16t^2+64t+6

The first part of this question asks us the maximum height of the ball.  From Physics, we learn that the maximum height of a projectile is reached when the velocity is 0, which happens to be right where the projectile stops for a nanosecond in the air to turn around and come back down.  We set the velocity function equal to 0 and solve for t.

0 = -32t + 64 and

0 = -32(t - 2).  By the Zero Product Property, either -32 = 0 or t - 2 = 0.  It's obvious that -32 does not equal 0, so t - 2 must equal 0.  Solving this for t:

t - 2 = 0 so

t = 2 seconds.  Since the maximum height is reached at a time of 2 seconds, we plug 2 seconds into the position function to get its position at 2 seconds (which is also the max height of the ball).

s(2)=-16(2)^2+64(2)+6 and

s(2) = -64 + 128 + 6 so

s(2) = 70 feet

Now we want to know when the ball will hit the ground.  "When" is a time value, and we know that the height of the ball on the ground is 0, so we sub in a 0 for s(t) and factor the quadratic.

Using the quadratic formula:

t=\frac{-64+/-\sqrt{4096-4(-16)(6)} }{-32} and

t=\frac{-64+/-\sqrt{4480} }{-32} which gives us the 2 solutions

t=\frac{-64+\sqrt{4480} }{-32} and

t=\frac{-64-\sqrt{4480} }{-32}

Plugging into your calculator, the first t = -.0916500 and the second t = 4.091

We all know that time cannot ever be negative, so our t value is 4.09.

Again, from Physics, we know that a projectile reaches it max height at halfway through its travels, so it just goes to follow logically that if it halfway through its travels at 2 seconds, then it will hit the ground at 4 seconds.  And it does!! How awesome is that?!

4 0
3 years ago
Expand & simplify 4(t+2)+6(t-4)
kondaur [170]

Answer:

10t - 16

Step-by-step explanation:

4(t + 2)+ 6(t - 4)

4t + 8 + 6t -24

10t - 16

8 0
3 years ago
Read 2 more answers
Fifteen more than half a number is 9
rusak2 [61]

For this case, the first thing we must do is define a variable.

We have then:

x: unknown number

We now write the equation that models the problem:

(\frac{1}{2}) x + 15 = 9

From here, we clear the value of x.

We multiply both sides of the equation by 2:

(\frac{2}{2}) x + 2 (15) = 2 (9)\\x + 30 = 18

We subtract 30 on both sides of the equation:

x + 30 - 30 = 18 - 30\\x = -12

Answer:

The value of the unknown number is given by:

x = -12

6 0
3 years ago
Read 2 more answers
Solve for x please!
Elina [12.6K]

x^2 = 81/100

Let's take the square root of each side

x = 0.9 and -0.9

3 0
3 years ago
Samir bought 3 pounds of cement to repair the cracks in his sidewalk.Each crack needs to be filled with 2 ounces of cement. Will
drek231 [11]

Answer: He will divide to convert pounds to ounces.

Step-by-step explanation:

7 0
3 years ago
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