in her last computer game, Lucy scored 3×10⁷ points. The first time she tried the game, she scored 6×10³ points. How many times as many points is Lucy's last score as her first score?​

Answer:

C. F(x) = x^2 + 4

Step-by-step explanation:

Took the test

Answer:

3

Step-by-step explanation:

3 times 30 = 90 then divide 90 and 30 you get 3

Answer:

2(3a-z)

cannot be further simplified

Step-by-step explanation:

Answer:

the answer will be [-3/4,1/2]

Step-by-step explanation:

Answer:10

Step-by-step explanation:

im guessing 800 50X4 = 200 and 200X4 = 800

i never learned this but i hope its correct.

Answer:

See below for Part A.

Part B)

[tex]\displaystyle h=\Big(\frac{125}{\pi}+27\Big)^\frac{2}{3}-9\approx7.4614[/tex]

Step-by-step explanation:

Part A)

The parabola given by the equation:

[tex]y^2=4ax[/tex]

From 0 to h is revolved about the x-axis.

We can take the principal square root of both sides to acquire our function:

[tex]y=f(x)=\sqrt{4ax}[/tex]

Please refer to the attachment below for the sketch.

The area of a surface of revolution is given by:

[tex]\displaystyle S=2\pi\int_{a}^{b}r(x)\sqrt{1+\big[f^\prime(x)]^2} \,dx[/tex]

Where r(x) is the distance between f and the axis of revolution.

From the sketch, we can see that the distance between f and the AoR is simply our equation y. Hence:

[tex]r(x)=y(x)=\sqrt{4ax}[/tex]

Now, we will need to find f’(x). We know that:

[tex]f(x)=\sqrt{4ax}[/tex]

Then by the chain rule, f’(x) is:

[tex]\displaystyle f^\prime(x)=\frac{1}{2\sqrt{4ax}}\cdot4a=\frac{2a}{\sqrt{4ax}}[/tex]

For our limits of integration, we are going from 0 to h.

Hence, our integral becomes:

[tex]\displaystyle S=2\pi\int_{0}^{h}(\sqrt{4ax})\sqrt{1+\Big(\frac{2a}{\sqrt{4ax}}\Big)^2}\, dx[/tex]

Simplify:

[tex]\displaystyle S=2\pi\int_{0}^{h}\sqrt{4ax}\Big(\sqrt{1+\frac{4a^2}{4ax}}\Big)\,dx[/tex]

Combine roots;

[tex]\displaystyle S=2\pi\int_{0}^{h}\sqrt{4ax\Big(1+\frac{4a^2}{4ax}\Big)}\,dx[/tex]

Simplify:

[tex]\displaystyle S=2\pi\int_{0}^{h}\sqrt{4ax+4a^2}\, dx[/tex]

Integrate. We can consider using u-substitution. We will let:

[tex]u=4ax+4a^2\text{ then } du=4a\, dx[/tex]

We also need to change our limits of integration. So:

[tex]u=4a(0)+4a^2=4a^2\text{ and } \\ u=4a(h)+4a^2=4ah+4a^2[/tex]

Hence, our new integral is:

[tex]\displaystyle S=2\pi\int_{4a^2}^{4ah+4a^2}\sqrt{u}\, \Big(\frac{1}{4a}\Big)du[/tex]

Simplify and integrate:

[tex]\displaystyle S=\frac{\pi}{2a}\Big[\,\frac{2}{3}u^{\frac{3}{2}}\Big|^{4ah+4a^2}_{4a^2}\Big][/tex]

Simplify:

[tex]\displaystyle S=\frac{\pi}{3a}\Big[\, u^\frac{3}{2}\Big|^{4ah+4a^2}_{4a^2}\Big][/tex]

FTC:

[tex]\displaystyle S=\frac{\pi}{3a}\Big[(4ah+4a^2)^\frac{3}{2}-(4a^2)^\frac{3}{2}\Big][/tex]

Simplify each term. For the first term, we have:

[tex]\displaystyle (4ah+4a^2)^\frac{3}{2}[/tex]

We can factor out the 4a:

[tex]\displaystyle =(4a)^\frac{3}{2}(h+a)^\frac{3}{2}[/tex]

Simplify:

[tex]\displaystyle =8a^\frac{3}{2}(h+a)^\frac{3}{2}[/tex]

For the second term, we have:

[tex]\displaystyle (4a^2)^\frac{3}{2}[/tex]

Simplify:

[tex]\displaystyle =(2a)^3[/tex]

Hence:

[tex]\displaystyle =8a^3[/tex]

Thus, our equation becomes:

[tex]\displaystyle S=\frac{\pi}{3a}\Big[8a^\frac{3}{2}(h+a)^\frac{3}{2}-8a^3\Big][/tex]

We can factor out an 8a^(3/2). Hence:

[tex]\displaystyle S=\frac{\pi}{3a}(8a^\frac{3}{2})\Big[(h+a)^\frac{3}{2}-a^\frac{3}{2}\Big][/tex]

Simplify:

[tex]\displaystyle S=\frac{8\pi}{3}\sqrt{a}\Big[(h+a)^\frac{3}{2}-a^\frac{3}{2}\Big][/tex]

Hence, we have verified the surface area generated by the function.

Part B)

We have:

[tex]y^2=36x[/tex]

We can rewrite this as:

[tex]y^2=4(9)x[/tex]

Hence, a=9.

The surface area is 1000. So, S=1000.

Therefore, with our equation:

[tex]\displaystyle S=\frac{8\pi}{3}\sqrt{a}\Big[(h+a)^\frac{3}{2}-a^\frac{3}{2}\Big][/tex]

We can write:

[tex]\displaystyle 1000=\frac{8\pi}{3}\sqrt{9}\Big[(h+9)^\frac{3}{2}-9^\frac{3}{2}\Big][/tex]

Solve for h. Simplify:

[tex]\displaystyle 1000=8\pi\Big[(h+9)^\frac{3}{2}-27\Big][/tex]

Divide both sides by 8π:

[tex]\displaystyle \frac{125}{\pi}=(h+9)^\frac{3}{2}-27[/tex]

Isolate term:

[tex]\displaystyle \frac{125}{\pi}+27=(h+9)^\frac{3}{2}[/tex]

Raise both sides to 2/3:

[tex]\displaystyle \Big(\frac{125}{\pi}+27\Big)^\frac{2}{3}=h+9[/tex]

Hence, the value of h is:

[tex]\displaystyle h=\Big(\frac{125}{\pi}+27\Big)^\frac{2}{3}-9\approx7.4614[/tex]

You seem to have left out that 0 ≤ x ≤ h.

From y² = 4ax, we get that the top half of the parabola (the part that lies in the first quadrant above the x-axis) is given by y = √(4ax) = 2√(ax). Then the area of the surface obtained by revolving this curve between x = 0 and x = h about the x-axis is

[tex]2\pi\displaystyle\int_0^h y(x) \sqrt{1+\left(\frac{\mathrm dy(x)}{\mathrm dx}\right)^2}\,\mathrm dx[/tex]

We have

y(x) = 2√(ax)   →   y'(x) = 2 • a/(2√(ax)) = √(a/x)

so the integral is

[tex]4\sqrt a\pi\displaystyle\int_0^h \sqrt x \sqrt{1+\frac ax}\,\mathrm dx[/tex]

[tex]=\displaystyle4\sqrt a\pi\int_0^h (x+a)^{\frac12}\,\mathrm dx[/tex]

[tex]=4\sqrt a\pi\left[\dfrac23(x+a)^{\frac32}\right]_0^h[/tex]

[tex]=\dfrac{8\pi\sqrt a}3\left((h+a)^{\frac32}-a^{\frac32}\right)[/tex]

Now, if y² = 36x, then a = 9. So if the area is 1000, solve for h :

[tex]1000=8\pi\left((h+9)^{\frac32}-27\right)[/tex]

[tex]\dfrac{125}\pi=(h+9)^{\frac32}-27[/tex]

[tex]\dfrac{125+27\pi}\pi=(h+9)^{\frac32}[/tex]

[tex]\left(\dfrac{125+27\pi}\pi\right)^{\frac23}=h+9[/tex]

[tex]\boxed{h=\left(\dfrac{125+27\pi}\pi\right)^{\frac23}-9}[/tex]

Answer:

$1

Step-by-step explanation:

Commission earned on sale = 2%

Total sales made = $50

Commission earned for this sale :

Commission percent * worth of sales made

0.02 * $50

= $1

Answer:

54

Step-by-step explanation:

Answer:

A) y=3x-4

Step-by-step explanation:

[tex]\sqrt{9a^{2}}=\sqrt{9}\sqrt{a^{2}}=3|a|[/tex]

As [tex]a < 0[/tex], [tex]|a|=-a[/tex], so [tex]3|a|=\boxed{-3a}[/tex]

Answer:

40,70,90

Step-by-step explanation:

9514 1404 393

Answer:

  x = 3.6

Step-by-step explanation:

The angle bisector divides the triangle proportionally.

The base segment to the right of the bisector is 6-x, so we can write the proportion ...

  x/6 = (6-x)/4

Multiplying by 12, we get ...

  2x = 3(6 -x)

  5x = 18 . . . . . add 3x

  x = 18/5 . . . .  divide by 5

  x = 3.6

_____

Alternate solution

Since the base of the triangle is given as a left-part and a sum-of-both-parts, we can write the proportion the same way:

  left-part / sum-of-both-parts = x/6 = 6/(6+4)

Then the solution is x = 6(6/10) = 36/10 = 3.6.

Doing it this way avoids having x on both sides of the equation, so makes solving the equation be "one step."

Answer:

expression 3

because expression 1 = 1

and expression 2 = 7

Answer:

their down payment is 30%

Step-by-step explanation:

Answer:

Step-by-step explanation:

if the price of the apartment is 140,000 dollars

lets find out how much is 1%

140,000/100=1400

42,000/1400=30  

42,000 is 30% of the apt price

49,000/1400=35

49,000 is 35% of the apt price

Answer:

1.5555556

1 5/9

14/9

Step-by-step explanation:

Answer:

x = -6 and x = 12 are parallel

Step-by-step explanation:

they are both vertical lines, one crosses the x-axis at -6, the other at 12

Answer:

y=5x-15, y=-1/5x-23/5

Step-by-step explanation:

Parallel lines have the same slope, so we know the line that is parallel to y=5x-3 must have slope 5.

We now have y=5x+b. We are given the point (2, -5) so substitute the x and y coordinates into the equation.

y=5x+b

-5=5(2)+b

-5=10+b

b=-15

So the equation of the parallel line is y=5x-15.

Perpendicular lines must have slopes that multiply to -1, so we know the line perpendicular to y=5x-3 must have slope -1/5.

We now have y=-1/5x+b. We are given the point (2, -5) so substitute the x and y coordinates into the equation.

y=-1/5x+b

-5=-1/5(2)+b

-5=-2/5+b

b=-23/5

So the equation of the perpendicular line is y=-1/5x-23/5.

Answer:

1.  y=3x-4

2. y=-3x+4

3. y= -1/3x+4

4. y= 1/3x-4

Step-by-step explanation:

1. slope= change in y/ change in x

  slope= -1-(-4)/1-0=3

  y=3x+b

 -4=3(0)+b

   b=-4

y=3x-4

2. slope=4-10/0-(-2)= -3

y=-3x+b

b=4

y=-3x+4

3. slope= -1/3

y=-1/3x+b

3= -1/3(3)+b

3= -1+b

b=4

y= -1/3x+4

4. y= 1/3x-4 (last one)

Answer:

n = 12

I hope this helps!

Answer:

i think its a beaner

Step-by-step explanation:

Step-by-step explanation:

3/5 of the book = 75 pages

5/5 of the book = 75 * (5/3) = 125 pages

Hence there are 125 pages in the whole book.

Answer:

Concept: Data Analysis

Given:

The equation of parallel line is [tex]y=-\dfrac{1}{4}x+5[/tex].

Required line passes through (2,-3).

To find:

The equation of line.

Solution:

We have,

[tex]y=-\dfrac{1}{4}x+5[/tex]

On comparing this equation with [tex]y=mx+b[/tex], where m is slope, we get

[tex]m=-\dfrac{1}{4}[/tex]

Slope of two parallel lines are always same. So, slope of required line is [tex]m=-\dfrac{1}{4}[/tex].

The required line passes through the point (2,-3) having slope [tex]m=-\dfrac{1}{4}[/tex], so the equation of line is

[tex]y-y_1=m(x-x_1)[/tex]

[tex]y-(-3)=-\dfrac{1}{4}(x-2)[/tex]

[tex]y+3=-\dfrac{1}{4}(x)-\dfrac{1}{4}(-2)[/tex]

[tex]y+3=-\dfrac{1}{4}(x)+\dfrac{1}{2}[/tex]

Subtracting 3 from both sides, we get

[tex]y=-\dfrac{1}{4}(x)+\dfrac{1}{2}-3[/tex]

[tex]y=-\dfrac{1}{4}(x)+\dfrac{1-6}{2}[/tex]

[tex]y=-\dfrac{1}{4}(x)-\dfrac{5}{2}[/tex]

Therefore, the equation of required line is [tex]y=-\dfrac{1}{4}(x)-\dfrac{5}{2}[/tex].

Answer: The price is now 14.40

Step-by-step explanation: The difference is 9.60

Answer:

B

Step-by-step explanation:

(3.2+4.8)+[20+19+(-4)]

8. +. 47. -4

=43

Answer:b

Step-by-step explanation:

Answer:

1.5 inches.

Step-by-step explanation:

A picture measuring 4 inches by 6 inches is placed inside a frame which has equal width around the entire picture.

The area of the frame and the picture is 63 square inches.