Find the absolute maximum and minimum values of the function f(x, y) = x2 + xy + y2 on the disc x2 + y2 ≤ 1

Gio should first complete the square by adding and subtracting [tex]\((\frac{3}{2})^2\)[/tex], leading to the equation [tex]\((x + \frac{3}{2})^2 = \frac{25}{4}\).[/tex]

To solve the quadratic equation [tex]\(5x^2 + 15x - 4 = 0\)[/tex] by completing the square, Gio should follow these steps:

1. Make the Coefficient of [tex]\(x^2\) 1:[/tex]

Divide the entire equation by the coefficient of [tex]\(x^2\)[/tex] to make it equal to 1. In this case, divide the equation by 5:

[tex]\[x^2 + 3x - \frac{4}{5} = 0.\][/tex]

2. Move the Constant to the Other Side:

Move the constant term to the other side of the equation:

[tex]\[x^2 + 3x = \frac{4}{5}.\][/tex]

3. Complete the Square:

  Add and subtract [tex]\((\frac{3}{2})^2\)[/tex] inside the parentheses to complete the square:

 [tex]\[x^2 + 3x + (\frac{3}{2})^2 = \frac{4}{5} + (\frac{3}{2})^2.\][/tex]

4. Factor the Perfect Square Trinomial:

  Factor the left side, which now is a perfect square trinomial:

 [tex]\[(x + \frac{3}{2})^2 = \frac{25}{4}.\][/tex]

5. Solve for (x):

  Take the square root of both sides and solve for (x):

[tex]\[x + \frac{3}{2} = \pm \frac{5}{2}.\][/tex]

6. Isolate (x):

  Isolate (x) on one side to find the solutions:

  [tex]\[x = -\frac{3}{2} \pm \frac{5}{2}.\][/tex]

Therefore, Gio should first complete the square by adding and subtracting [tex]\((\frac{3}{2})^2\)[/tex], leading to the equation [tex]\((x + \frac{3}{2})^2 = \frac{25}{4}\).[/tex]

The equation that can be used to determine how long after Car X leaves Car Y will catch up is D. 55x = 60(x - 1).

To determine when Car Y will catch up to Car X, we need to find the time it takes for Car Y to cover the same distance as Car X. Since Car Y leaves 1 hour later, it will travel for (x - 1) hours, where x represents the time Car X has been traveling. So, the equation would be:

D. 55x = 60(x - 1)

This equation accounts for the distance traveled by both cars over the same time frame, considering Car Y started an hour later.

Complete question : Car X leaves Northtown traveling at a steady rate of 55 mph. Car Y leaves 1 hour later following Car X, traveling at a steady rate of 60 mph. Which equation can be used to determine how long after Car X leaves Car Y will catch up? A. 55x=60x-1 B.55x=60x C.60x=55x(x-1) D.55x=60(x-1).

The positive value of the discriminant implies that there are two real unequal roots and this can be determined by using the formula of the discriminant.

Given :

The quadratic equation is [tex](0=-2x^2+3)[/tex].

The following steps can be used in order to determine the total number of real solutions does the equation have:

Step 1 - Write the given quadratic equation.

[tex]0=-2x^2+0x+3[/tex]

Step 2 - The formula of discriminant is given below:

[tex]D = b^2-4ac[/tex]

where 'b' is the coefficient of 'x', 'a' is the coefficient of [tex]x^2[/tex], and 'c' is the constant.

Step 3 - Now, substitute the values of a, b, and c in the above formula.

[tex]D = (0)^2-4\times (-2)\times 3[/tex]

[tex]D = 24[/tex]

Step 4 - The positive value of the discriminant implies that there are two real unequal roots.

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To solve this, we use Pythagoras' theorem as we have a right-angled triangle formed by the building, its shadow and the line between the top of the building to the end of the shadow. We find that the length of the shadow is approximately 16.7 meters.

To solve this problem, we need to apply Pythagoras' theorem because the building, the shadow and the line between the top of the building to the end of the shadow form a right-angled triangle.  

Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as:

a² + b² = c²

In this scenario, the height of the building is one side of the triangle (a), the shadow is the other side (b), and the line from the top of the building to the tip of the shadow is the hypotenuse (c).

So, to find the length of the shadow, we rearrange the formula to solve for b:

b = sqrt(c² - a²)

Then, we substitute the known values:

b = sqrt( (37m)² - (33m)²) = sqrt(1369 - 1089) = sqrt(280) = 16.7m

So, the length of the shadow is approximately 16.7 meters.

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To make 30 cupcakes Margaret needs 4.375 cups of vanilla.

The ratio is the division of the two numbers.

For example, a/b, where a will be the numerator and b will be the denominator.

Proportion is the relation of a variable with another. It could be direct or inverse.

Given that,

Margaret uses 1 3/4 teaspoon of vanilla to make 12 cupcakes

So,

1 3/4 teaspoon vanilla ⇒ 12 cupcakes

(4×1 + 3)/4 teaspoon ⇒ 12 cupcakes

7/4 teaspoon vanilla ⇒  12 cupcakes

Divide by 12 into both sides,

7/(4×12)  teaspoon vanilla ⇒ 1 cupcake

Now multiply by 30 on both sides,

(7 × 30)/(4 × 12)teaspoon vanilla ⇒ 30 cupcake

30 cupcakes ⇒ 4.375 teaspoon vanilla.

Hence "To make 30 cupcakes Margaret needs 4.375 cups of vanilla".

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Answer-

[tex]\frac{\log (x+2)}{\log 4}[/tex] is the correct answer.

Solution-

The formula for change of base in logarithm is given by,

[tex]\log_{b}a=\frac{\log_{c}a}{\log_{c}b}[/tex]

If we take the common base as 'e' , the formula becomes,

[tex]\log_{b}a=\frac{\log a}{\log b}[/tex]

Applying this formula,

[tex]\log_{4}(x+2)=\frac{\log (x+2)}{\log 4}[/tex]

∴ As the first option matches with our answer, hence it is the correct answer.

( In the fourth option a bracket is missing, so it is not the correct option)

Compound interest is the interest generated overtime on a sum of money invested. The time it will take for the amount of money to double is 5.9 years

Compound interest is the interest generated overtime on a sum of money invested. The formula for calculating compound interest is expressed as:
V(t) = P(1+r/n)^nt

where:

Given the following

V(t) = 2P
r = 0.12
n = 4

Substitute
2P = P(1+0.12/4)^4t
2 =  (1.03)^4t
ln2 = 4t ln(1.03)
0.6931 = 0.1182t
t = 0.6931/0.1182
t = 5.9 years

Hence the time it will take for the amount of money to double is 5.9 years

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