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Word Knowledge

Archetypal most nearly means:

classic

Characteristic of the original model or perfect example.

Math Knowledge

Solve for x:
x² - 5x - 6 = 0

-1 or 6

The first step to solve a quadratic equation that's set to zero is to factor the quadratic equation:

x² - 5x - 6 = 0
(x + 1)(x - 6) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (x + 1) or (x - 6) must equal zero:

If (x + 1) = 0, x must equal -1
If (x - 6) = 0, x must equal 6

So the solution is that x = -1 or 6

Mechanical Comprehension

If A = 9 ft. and the green box weighs 30 lbs. what is the torque acting on the A side of this lever?

270 ft⋅lb

For a lever, torque is weight x distance from the fulcrum which, in this case, is: 30 ft. x 9 lbs. = 270 ft⋅lb

Arithmetic Reasoning

If a rectangle is twice as long as it is wide and has a perimeter of 12 meters, what is the area of the rectangle?

8 m²

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 12 meters so the equation becomes: 2w + 2h = 12.

Putting these two equations together and solving for width (w):

2w + 2h = 12
w + h = \( \frac{12}{2} \)
w + h = 6
w = 6 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 6 - 2w
3w = 6
w = \( \frac{6}{3} \)
w = 2

Since h = 2w that makes h = (2 x 2) = 4 and the area = h x w = 2 x 4 = 8 m²

Arithmetic Reasoning

Solve 2 + (5 + 2) ÷ 5 x 3 - 4²

-9\(\frac{4}{5}\)

Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

2 + (5 + 2) ÷ 5 x 3 - 4²
P: 2 + (7) ÷ 5 x 3 - 4²
E: 2 + 7 ÷ 5 x 3 - 16
MD: 2 + \( \frac{7}{5} \) x 3 - 16
MD: 2 + \( \frac{21}{5} \) - 16
AS: \( \frac{10}{5} \) + \( \frac{21}{5} \) - 16
AS: \( \frac{31}{5} \) - 16
AS: \( \frac{31 - 80}{5} \)
\( \frac{-49}{5} \)
-9\(\frac{4}{5}\)

Arithmetic Reasoning

If \( \left|x + 9\right| \) - 3 = 7, which of these is a possible value for x?

-19

First, solve for \( \left|x + 9\right| \):

The value inside the absolute value brackets can be either positive or negative so (x + 9) must equal + 10 or -10 for \( \left|x + 9\right| \) to equal 10:

x + 9 = 10
x = 10 - 9
x = 1

x + 9 = -10
x = -10 - 9
x = -19

So, x = -19 or x = 1.

Word Knowledge

Allison's painting met with plaudits from the critics, who had dismissed her earlier work.

kudos

Enthusiastic approval.

Word Knowledge

"Having cake now will preclude dessert later," said Amber's mother.

prevent

To rule out in advance.

Electronics Information

The diameter of a number 12 wire is __________ the diameter of a number 6 wire?

less than

The larger the number the smaller the diameter of the wire. So, the diameter of a number 12 wire is less than the diameter of a number 6 wire.

Automotive Information

Exhaust gases from cylinder exhaust valves are collected by the:

exhaust manifolds

The cast iron exhaust manifolds collect engine exhaust gas from multiple cylinder exhaust valves and deliver it to the exhaust pipe. Exhaust manifolds can be generic or specially tuned (header pipes) to the engine. Header pipes deliver higher performance but are more expensive and less durable.