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9 days ago

Complete each statement in the steps to solve x2 – 6x – 7 = 0 using the process of completing the square. Isolate the constant b

y both sides of the equation. Add to both sides of x2 – 6x = 7 to form a perfect square trinomial while keeping the equation balanced. Write the trinomial x2 – 6x + 9 as squared. Use the square root property of equality to get x – 3 = ± . Isolate the variable to get solutions of –1 and 7.
ANSWERS:
Adding 7 to
9
x-3
4

Mathematics

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Jane is training for a triathlon. After swimming a few laps, she leaves the health club and bikes 16 miles south. She then runs

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Answer:

The formula to determine the distance Jane's trainer bikes is $x=\sqrt{16^2+12^2}$ .

Detailed explanation:

A diagram has been included for clarity.

Known values:

Distance biked to the south = 16 miles

Distance ran west = 12 miles

We need to determine the distance covered by Jane's trainer.

Solution:

Let the biking distance be represented by 'x'.

We will assume it forms a right-angled triangle.

According to the Pythagorean theorem, it states that;

"The square of the hypotenuse is equal to the sum of the squares of the other two sides."

When set in equation form, this gives us;

$x^2=16^2+12^2\\\\x=\sqrt{16^2+12^2}$

Thus, the equation representing the distance biked by Jane's trainer is $x=\sqrt{16^2+12^2}$ .

Upon solving, we find;

$x=\sqrt{16^2+12^2}\\\\x= \sqrt{256+144}\\ \\x=\sqrt{400} \\\\x=20\ miles$

Consequently, Jane's trainer covers a distance of 20 miles.

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1 month ago

If 30,000 cm2 of material is available to make a box with a square base and an open top, what is the largest possible volume (in

Inessa [12570]

Answer:

The highest achievable volume for the box is 2000000 cubic meters.

Step-by-step explanation:

Below is an outline of the volume ( $V$ ), measured in cubic centimeters, and surface area ( $A_{s}$ ), measured in square centimeters, for a box featuring a square base:

$A_{s} = l^{2}+h\cdot l$ (1)

$V = l^{2}\cdot h$ (2)

Where:

$l$ - The length of the base's side, in centimeters.

$h$ - The height of the box, in centimeters.

Using (2), we isolate $h$ in the formula:

$h = \frac{V}{l^{2}}$

Then, we substitute into (1) and simplify the outcome:

$A_{s} = l^{2}+ \frac{V}{l}$

$A_{s}\cdot l = l^{3}+V$

$V = A_{s}\cdot l -l^{3}$ (3)

Next, we calculate the first and second derivatives of this expression:

$V' = A_{s}-3\cdot l^{2}$ (4)

$V'' = -6\cdot l$ (5)

If $V' = 0$ and $A_{s} = 30000\,cm^{2}$ , then we find that the critical value for the base's side length is:

$30000-3\cdot l^{2} = 0$

$3\cdot l^{2} = 30000$

$l = 100\,cm$

Subsequently, we assess this outcome using the second derivative's expression:

$V'' = -600$

According to Second Derivative Test, this critical value signifies an absolute maximum. Consequently, the largest volume obtainable for the box is:

$V = 30000\cdot l - l^{3}$

$V = 2000000\,cm^{3}$

The highest achievable volume for the box is 2000000 cubic meters.

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For the angles α and β in the figures, find cos(α + β)?

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Answer:

Step-by-step explanation:

The total count of laptops, N = 20

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Case 1:

Both purchased laptops are non-defective

Probability of selecting 2 non-defective laptops

The count of non-defective laptops equals 20 - 3 = 17

Overall laptop count = 20

Probability, P = 17/20 = 0.85

Case 2:

One laptop is found defective

The probability of choosing 1 defective and 1 non-defective laptop

P' = 3/20 x 17/19 = 0.134

Case 3:

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