Simplify: |72 - 9 • 7| - 8|16

How are quadratic equations different from linear equations?

worty [1.4K]

The fundamental difference between the linear and quadratic functions is observed in their respective graphs. A linear function is represented by a straight line and a quadratic function is represented by a curve called a parabola.

See file attached.

3 0

3 years ago

Diagram 5 shows a right cylinder with a diameter of 2xcm. Given that the total surface area of the cylinder is 96cm³.Find the ma

Paraphin [41]

Given:

The diameter of the right cylinder is 2x cm.

The total surface area is 96 cm cube.

The radius is calculated as,

$\begin{gathered} r=\frac{d}{2} \\ r=\frac{2x}{2} \\ r=x\text{ cm} \end{gathered}$

The total surface area is,

$\begin{gathered} S=2\pi rh+2\pi(r)^2 \\ 96=2\pi xh+2\pi(x^2) \\ h=\frac{96-2\pi(x^2)}{2\pi x} \end{gathered}$

Volume is,

$\begin{gathered} V=\pi(r)^2h \\ =\pi(x^2)\frac{96-2\pi(x^2)}{2\pi x} \\ =\frac{x(96-2\pi(x^2)}{2} \end{gathered}$

Now, differentiate with respect to x,

$\begin{gathered} \frac{dV}{dx}^{}=\frac{d}{dx}(\frac{x(96-2\pi(x^2)}{2}) \\ =\frac{d}{dx}\mleft(x\mleft(-\pi x^2+48\mright)\mright) \\ =\frac{d}{dx}\mleft(x\mright)\mleft(-\pi x^2+48\mright)+\frac{d}{dx}\mleft(-\pi x^2+48\mright)x \\ =1\cdot\mleft(-\pi x^2+48\mright)+\mleft(-2\pi x\mright)x \\ =84-3\pi(x^2)\ldots\ldots\ldots\ldots\text{.}(1) \end{gathered}$

Now,

$\begin{gathered} \frac{dV}{dx}=0 \\ 84-3\pi(x^2)=0 \\ x^2=\frac{16}{\pi} \\ x=\sqrt[]{\frac{16}{\pi}} \end{gathered}$

Now, differentiate (1) with respect to x again,

$\begin{gathered} \frac{d^2V}{dx^2}=\frac{d}{dx}(84-3\pi(x^2)) \\ =-6\pi x \\ At\text{ x=}\sqrt[]{\frac{16}{\pi}} \\ \frac{d^2V}{dx^2}=-6\pi\sqrt[]{\frac{16}{\pi}}$

Since, the double derivative is negative.

$So,\text{ the volume is maximum at }\sqrt[]{\frac{16}{\pi}}$

So, the volume becomes,

$\begin{gathered} V=\pi(x^2)h \\ V=\pi(\sqrt[]{\frac{16}{\pi}})^2h \\ V=\frac{16h}{\pi} \end{gathered}$

Answer: maximum volume of the cylinder is,

6 0

1 year ago

What is the measure of the central angle.

Nana76 [90]

Answer:

35*

Step-by-step explanation:

IT IS THE SAME ANGLE

3 0

3 years ago

Read 2 more answers

Summary

mina [271]

Answer:

i!^dkk

Step-by-step explanation:

more than

3 0

2 years ago

Read 2 more answers

A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum ar

Artyom0805 [142]

The farmer is going to construct a rectangular shaped barn with dimensions

a and b,

such that 2a+2b=2020, (the perimeter is 2 width+2 length)

dividing by 2, we have a+b=1010

writing b in terms of a, we have b=1010-a

so f(a)=a(1010-a) is the function which calculates the area depending on a.

for example the area enclosed is the dimensions are a= 10, b=1000

is f(10)=10*(1010-10)=10*1000=10,000 square feet.

f(a) is a polynomial function of degree 2, that is a quadratic.

The graph of a quadratic polynomial is a parabola, in our case, a parabola which opens downward since the sign of the coefficient of the largest term of the quadratic, is negative.

since the quadratic is given in factorized form, we easily find that the roots are a=0 and a=1010

since f(0)=0 and f(1010)=0.

in the middle of these points, we have 1010/2=505, which is the x-coordinate of the vertex.

f(505) gives us the highest point, which is f(505)=505(1010-505)= $505^{2}$ =255,025 ft squared.

the highest point of the parabola, represents the largest value of f so the largest possible area.

Answer: 255,025 ft squared.

3 0

2 years ago

Simplify: |72 - 9 • 7| - 8|16 - 2|