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KatRina [158]
3 years ago
13

Consider the table below that represents a quadratic function.

Mathematics
1 answer:
zlopas [31]3 years ago
6 0
It would be -1/2 because the formula to find slope (rate of change) is (y2-y1)/(x2-x1) and in this case it would be (4.5-2)/(-3--2) which simplifies to (2.5)/(-5) which is -1/2
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Find the distance from the point (1,4) to the line y = 1/3x - 3
Troyanec [42]

Answer:

Step-by-step explanation:

If I'm not mistaken, and I very well could be, this is a calculus problem(?). In order to find the distance without calculus you'd need a point on the given line to use to find the distance in the distance formula. But you don't have a point on the given line, so we can find the shortest distance between the point (1, 4) and the given line using the derivative of the polynomial formed when using the distance formula.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} and we have the x and y for x2 (or x1...it doesn't matter which you choose to fill in):

d=\sqrt{(1-x)^2+(4-y)^2}

but what we find is that we have too many unknowns here, namely, the distance, the x coordinate, and the y coordinate. So we can replace the y coordinate with what y is equal to in terms of the linear equation:

d=\sqrt{(1-x)^2+(4-\frac{1}{3}x-3)^2 } and simplify:

d=\sqrt{(1-x)^2+(7-\frac{1}{3}x)^2 }

. No we'll expand each binomial by squaring:

d=\sqrt{(1-2x+x^2)+(49-\frac{14}{3}x+\frac{1}{9}x^2)  }

.  Combining like terms gives us

d=\sqrt{\frac{10}{9}x^2-\frac{20}{3}x+50  }

The distance between the point (1, 4) and the given line will be at a minimum when the polynomial above is at a minimum. We find the value of x for which the polynomial is at a minimum by finding its derivative, setting the derivative equal to 0, and then solving for x. The derivative of the polynomial is

\frac{20}{9}x-\frac{20}{3}

Setting equal to 0 and getting rid of the denominators gives us

20x - 60 = 0

Solving for x gives us

20x = 60 and x = 3.

That's the value of x that gives us the shortest distance between (1, 4) and the line y = 1/3x - 3. Sub into the distance formula that x value to find the distance:

d=\sqrt{(\frac{10}{9})(3)^2-(\frac{20}{3})(3)+50   }

which simplifies down, finally, to

x ≈ 6.325 units

8 0
3 years ago
The coordinates of point A on a grid are (−2, −4). Point A is reflected across the y-axis to obtain point B. The coordinates of
Trava [24]

Answer:

Search up

Step-by-step explanation:

4 0
3 years ago
Convert the decimal to a fraction in simplest form:
Shkiper50 [21]

Answer:

Decimal 0.333 to a fraction in simplest form is:   \frac{333}{1000}

Step-by-step explanation:

Given the decimal

0.333

Multiply and divide by 10 for every number after the decimal point.

There are three digits to the right of the decimal point, therefore multiply and divide by 1000.

Thus,

0.333=\frac{0.333\cdot \:\:1000}{1000}

         =\frac{333}{1000}         ∵ 0.333×1000 = 333

Let us check if we can reduce the fraction \frac{333}{1000}

For this, we need to find a common factor of 333 and 1000 in order to cancel it out.

But, first, we need to find the Greatest Common Divisor (GCD) of 333, 1000

<u>Greatest Common Divisor (GCD) : </u>

The GCD of a, b is the largest positive number that divides both a and b without a remainder.

Prime Factorization of 333:      3 · 3 · 37

Prime Factorization of 1000:      2 · 2 · 2 · 5 · 5 · 5

As there is no common factor for 333 and 1000, therefore, the GCD is 1.

Important Tip:

  • As GCD is 1, therefore the fraction can not be simplified.

Therefore, decimal 0.333 to a fraction in simplest form is:   \frac{333}{1000}

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Which part of a sphere is the same distance from all points that are on the sphere?
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D) Center I'm sure of it
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