Solve for x.<br> Your answer must be simplified. −30< x/-4

Log3 (x squared + 7x + 21) = 2

Nezavi [6.7K]

Step-by-step explanation:

log <base a> b = x

means

a^x = b

So

3^2 = x^2+7x+21

x^2 + 7x + 21 - 9 = 0

x^2 + 7x + 12 = 0

(x+3)(x+4)

x = -3 or -4

3 0

3 years ago

What is the area of a circle with a radius of 6 inches? (Leave your answer in terms of Pi. This means: do not multiple by 3.14,

pickupchik [31]

Answer:

36π in^2

Step-by-step explanation:

Area of a circle = π(6)^2 = 36π

6 0

2 years ago

How come lemon washing up liquid contains real lemons, but lemon juice contains artificial flavorings.

attashe74 [19]

Answer:

Lemon juice is acidic and therefore is deemed unsafe to consume this can start to destroy the pancreas and other major arteries if not removed quickly

Step-by-step explanation:

3 0

3 years ago

A tutor work with 3 students. The oldest student is 6 years oldder then the middle student, and the youngest student is 3 years

Dennis_Churaev [7]

Answer:

9 years old.

Step-by-step explanation:

Since the ages of all of these students are being represented in relation to the middle student's age then we can create formulas for each student and then apply them to the formula for the sum of their ages and solve for the middle child's age (M) like so... (oldest child's age is represented by variable O and youngest child's age is represented by variable Y)

O = M + 6

Y = M - 3

30 = (M+6) + M + (M-3) ... combine like terms

30 = 3M + 3 ... subtract 3 from both sides

27 = 3M ... divide both sides by 3

9 = M

Now we know that the middle child is 9 years old.

6 0

2 years ago

A company wishes to manufacture some boxes out of card. The boxes will have 6 sides (i.e. they covered at the top). They wish th

Serhud [2]

Answer:

The dimensions are, base $b=\sqrt[3]{200}$ , depth $d=\sqrt[3]{200}$ and height $h=\sqrt[3]{200}$ .

Step-by-step explanation:

First we have to understand the problem, we have a box of unknown dimensions (base $b$ , depth $d$ and height $h$ ), and we want to optimize the used material in the box. We know the volume $V$ we want, how we want to optimize the card used in the box we need to minimize the Area $A$ of the box.

The equations are then, for Volume

$V=200cm^3 = b.h.d$

For Area

$A=2.b.h+2.d.h+2.b.d$

From the Volume equation we clear the variable $b$ to get,

$b=\frac{200}{d.h}$

And we replace this value into the Area equation to get,

$A=2.(\frac{200}{d.h} ).h+2.d.h+2.(\frac{200}{d.h} ).d$

$A=2.(\frac{200}{d} )+2.d.h+2.(\frac{200}{h} )$

So, we have our function $f(x,y)=A(d,h)$ , which we have to minimize. We apply the first partial derivative and equalize to zero to know the optimum point of the function, getting

$\frac{\partial A}{\partial d} =-\frac{400}{d^2}+2h=0$

$\frac{\partial A}{\partial h} =-\frac{400}{h^2}+2d=0$

After solving the system of equations, we get that the optimum point value is $d=\sqrt[3]{200}$ and $h=\sqrt[3]{200}$ , replacing this values into the equation of variable $b$ we get $b=\sqrt[3]{200}$ .

Now, we have to check with the hessian matrix if the value is a minimum,

The hessian matrix is defined as,

$H=\left[\begin{array}{ccc}\frac{\partial^2 A}{\partial d^2} &\frac{\partial^2 A}{\partial d \partial h}\\\frac{\partial^2 A}{\partial h \partial d}&\frac{\partial^2 A}{\partial p^2}\end{array}\right]$

we know that,

$\frac{\partial^2 A}{\partial d^2}=\frac{\partial}{\partial d}(-\frac{400}{d^2}+2h )=\frac{800}{d^3}$

$\frac{\partial^2 A}{\partial h^2}=\frac{\partial}{\partial h}(-\frac{400}{h^2}+2d )=\frac{800}{h^3}$

$\frac{\partial^2 A}{\partial d \partial h}=\frac{\partial^2 A}{\partial h \partial d}=\frac{\partial}{\partial h}(-\frac{400}{d^2}+2h )=2$

Then, our matrix is

$H=\left[\begin{array}{ccc}4&2\\2&4\end{array}\right]$

Now, we found the eigenvalues of the matrix as follow

$det(H-\lambda I)=det(\left[\begin{array}{ccc}4-\lambda&2\\2&4-\lambda\end{array}\right] )=(4-\lambda)^2-4=0$

Solving for $\lambda$ , we get that the eigenvalues are: $\lambda_1=2$ and $\lambda_2=6$ , how both are positive the Hessian matrix is positive definite which means that the function $A(d,h)$ is minimum at that point.

4 0

2 years ago

Solve for x. Your answer must be simplified. −30< x/-4