2. Quentin has 18 unit cubes.How many different rectangular prisms can he build if he uses all of the cubes? Please help

A roller coaster at an amusement park rises to a height of 196 feet above the ground. So far, it has risen 36% of this height. T

Eva8 [605]

Answer:

Step-by-step explanation:

196*36%=70.56 feet

He rounded the percent to 40% and the number to 200.

6 0

3 years ago

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1:Under what condition will the line px+py+r=0 mat be a normal to the circke x²+y²+2gx+2fy+c=0

ahrayia [7]

Answer:

The normal overlaps with the diameter, so it passes through the center.

Let's find the center of the circle:

x² + y² + 2gx + 2fy + c = 0
(x + g)² + (y + f)² = c + g² + f²

The center is:

(-g, -f)

Since the line passes through (-g, -f) the equation of the line becomes:

p(-g) + p(-f) + r = 0
r = p(g + f)

This is the required condition

Rewrite equations and find centers and radius of both circles.

Circle 1

x² + y² + 2ax + c² = 0
(x + a)² + y² = a² - c²
The center is (-a, 0) and radius is √(a² - c²)

Circle 2

x² + y² + 2by + c² = 0
x² + (y + b)² = b² - c²
The center is (0, -b) and radius is √(b² - c²)

The distance between two centers is same as sum of the radius of them:

d = √(a² + b²)

Sum of radiuses:

√(a² - c²) + √(b² - c²)

Since they are same we have:

√(a² + b²) = √(a² - c²) + √(b² - c²)

Square both sides:

a² + b² = a² - c² + b² - c² + 2√(a² - c²)(b² - c²)
2c² = 2√(a² - c²)(b² - c²)

Square both sides:

c⁴ = (a² - c²)(b² - c²)
c⁴ = a²b² - a²c² - b²c² + c⁴
a²c² + b²c² = a²b²

Divide both sides by a²b²c²:

1/a² + 1/b² = 1/c²

Proved

6 0

3 years ago

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What is the midpoint of EF If E has the coordinates of (2,4) and F has coordinates of (6,8)

allsm [11]

Answer:

$\boxed {\boxed {\sf (4,6) }}$

Step-by-step explanation:

The midpoint formula is:

$(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2} )$

where (x₁, y₁) and (x₂, y₂) are the points are the endpoints.

We are given the points: (2,4) and (6,8). Therefore,

$x_1=2\\y_1=4 \\x_2=6 \\y_2=8$

Substitute the values into the formula.

$(\frac{2+6}{2},\frac{4+8}{2})$

Find the x coordinate first.

Add 2 and 6: 2+6=8
Divide by 2: 8/2=4

$(4, \frac{4+8}{2})$

Find the y-coordinate next.

Add 4 and 8: 4+8= 12
Divide by 2: 12/2=6

$(4,6)$

The midpoint of EF is (4,6)

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3 years ago

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Preview Activity 3.5.1. A spherical balloon is being inflated at a constant rate of 20 cubic inches per second. How fast is the

ArbitrLikvidat [17]

Answer:

The radius is increasing at a rate of approximately 0.044 in/s when the diameter is 12 inches.

Because $\frac{dr}{dt}=\frac{5}{36\pi }>\frac{dr}{dt}=\frac{5}{64\pi }$ the radius is changing more rapidly when the diameter is 12 inches.

Step-by-step explanation:

Let $r$ be the radius, $d$ the diameter, and $V$ the volume of the spherical balloon.

We know $\frac{dV}{dt}=20 \:{in^3/s}$ and we want to find $\frac{dr}{dt}$

The volume of a spherical balloon is given by

$V=\frac{4}{3} \pi r^3$

Taking the derivative with respect of time of both sides gives

$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$

We now substitute the values we know and we solve for $\frac{dr}{dt}$ :

$d=2r\\\\r=\frac{d}{2}$

$r=\frac{12}{2}=6$

$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\\\\\frac{dr}{dt}=\frac{\frac{dV}{dt}}{4\pi r^2} \\\\\frac{dr}{dt}=\frac{20}{4\pi(6)^2 } =\frac{5}{36\pi }\approx 0.044$

The radius is increasing at a rate of approximately 0.044 in/s when the diameter is 12 inches.

When d = 16, r = 8 and $\frac{dr}{dt}$ is:

$\frac{dr}{dt}=\frac{20}{4\pi(8)^2}=\frac{5}{64\pi }\approx 0.025$

The radius is increasing at a rate of approximately 0.025 in/s when the diameter is 16 inches.

Because $\frac{dr}{dt}=\frac{5}{36\pi }>\frac{dr}{dt}=\frac{5}{64\pi }$ the radius is changing more rapidly when the diameter is 12 inches.

8 0

3 years ago

World grain production was 1241 million tons in 1975 and 2048 million tons in 2005, and has been increasing at an approximately

djyliett [7]

Answer:

The equation of line is $P(t)=26.9t+1241$

Step-by-step explanation:

Consider the provided information.

World grain production was 1241 million tons in 1975 and 2048 million tons in 2005,

We need to find the linear function.

The difference of the year is: 2005-1975=30

The function represents the world grain production at time t years after 1975, Thus, the first points is (0,1241) and the second points is (30,2048)

Find the slope of the line by using the formula: $m=\frac{y_2-y_1}{x_2-x_1}$

Substitute the respective values in the above formula.

$m=\dfrac{2048-1241}{30-0}\\\\m=\dfrac{807}{30}\\\\m=26.9$

The slope of the linear function is 26.9

The slope intercept form is: $y=mx+b$

Where b is the y intercept. y intercept is a point where the value of x is 0.

Therefore y intercept of the linear function is 1241 because the first points is (0,1241).

Hence, the equation of line is $P(t)=26.9t+1241$

7 0

4 years ago