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3 years ago

What is molecular gastronomy

Mathematics

1 answer:

MA_775_DIABLO [31]3 years ago

6 0

Answer:

Molecular gastronomy, or progressive cuisine, is a movement that incorporates science and new techniques in the preparation, transformation and artistic presentation of food. It is the study of molecules as they relate to the chemical and physical processes of cooking.

Step-by-step explanation:

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Determine all prime numbers a, b and c for which the expression a ^ 2 + b ^ 2 + c ^ 2 - 1 is a perfect square .

kogti [31]

Answer:

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Step-by-step explanation:

From Algebra we know that a second order polynomial is a perfect square if and only if $(x+y)^{2} = x^{2} + 2\cdot x\cdot y + y^{2}$ . From statement, we must fulfill the following identity:

$a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}$

By Associative and Commutative properties, we can reorganize the expression as follows:

$a^{2} + (b^{2}-1) + c^{2} = x^{2} + 2\cdot x \cdot y + y^{2}$ (1)

Then, we have the following system of equations:

$x = a$ (2)

$(b^{2}-1) = 2\cdot x\cdot y$ (3)

$y = c$ (4)

By (2) and (4) in (3), we have the following expression:

$(b^{2} - 1) = 2\cdot a \cdot c$

$b^{2} = 1 + 2\cdot a \cdot c$

$b = \sqrt{1 + 2\cdot a\cdot c}$

From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, $a, b, c > 1$ . If $a$ , $b$ and $c$ are prime numbers, then $2\cdot a\cdot c$ must be an even composite number, which means that $a$ and $c$ can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.

In addition, $b$ must be a natural number, which means that:

$1 + 2\cdot a\cdot c \ge 4$

$2\cdot a \cdot c \ge 3$

$a\cdot c \ge \frac{3}{2}$

But the lowest possible product made by two prime numbers is $2^{2} = 4$ . Hence, $a\cdot c \ge 4$ .

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Example: $a = 2$ , $c = 2$

$b = \sqrt{1 + 2\cdot (2)\cdot (2)}$

$b = 3$

4 0

3 years ago

Which equation of a line passes through the points (3, −1) and (6, 1)?

Vaselesa [24]

Answer:

Y = 2/3x - 3

Step-by-step explanation:

Recall the general equation for a straight line is

y = mx + b

where m is the gradient and b is the y-intercept

given 2 points whose coordinates are (x1, y1) and (x2, y2), m can be found with the following formula:

m = $\frac{y1-y2}{x1-x2}$

in this case, x1 = 3, y1 = -1, x2 = 6, y2=1

applying these values to the formula for m will give

m = (-1 -1) / (3-6) = 2/3

We can see immediately that only the first (top-most) answer has this value for m and we can guess that this is probably the answer.

But we can still check to confirm:

If we substitute this back into the general equation, we get:

y = (2/3)x + b

In order to find the value for b, we substitute any one of the 2 given points back into this equation. Lets choose (6,1)

1 = (2/3)(6) + b

1 = 4 + b

b = -3

Confirm Y = 2/3x - 3 is the answer.

4 0

3 years ago

The center of a circle is (4, 6), and an endpoint of a diameter is (2, 5). What is the other endpoint of the diameter?

valentinak56 [21]

Consider this option:
1. if the point (4;6) is the centre of the circle and the point (2;5) is the first endpoint of its diameter, then point (4;6) is the middle point of the diameter (it means that is the middle between the 1st and the 2d endpoints of diameter).
2. using the property described above:
for x of the 2d endpoint of the diameter: x=4*2-2=6;
for y of the 2d endpoint of the diameter: y=6*2-5=7.

answer: (6;7)

3 0

3 years ago

Read 2 more answers

Kate has a coin collection. She keeps seven of the coins in a box, which is only 5% of her entire collection. What is the total

tia_tia [17]

Answer:

D. 140

Step-by-step explanation:

8 0

3 years ago

Read 2 more answers

3x + y = 7 <br> 4x + 2y = 16