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# Mathematics

## Discover

• 0

### Finding Derivative of a Point?

g(x) = 4x² - 2x + 15 ← this is a function

g(5) = 100 - 10 + 15 = 105 → the representative curve of the function passes through (5 ; 105)

g'(x) = 8x - 2 ← this is its derivative

g'(5) = 40 - 2 = 38 ← this is in fact the slope of the tangent line to the curve at point (5 ; 105)

g(x) = 4x² - 2x + 15 ← this is the function → to calculate the derivative:

Lim [g(x₀ + h) - g(x₀)] / h

h → 0

Lim [ { 4.(x₀ + h)² - 2.(x₀ + h) + 15 } - { 4x₀² - 2x₀ + 15 } ] / h

h → 0

Lim [ { 4.(x₀² + 2x₀.h + h²) - 2x₀ - 2h + 15 } - 4x₀² + 2x₀ - 15 ] / h

h → 0

Lim [4x₀² + 8x₀.h + 4h² - 2x₀ - 2h + 15 - 4x₀² + 2x₀ - 15] / h

h → 0

Lim [8x₀.h + 4h² - 2h] / h

h → 0

Lim h.[8x₀ + 4h - 2] / h

h → 0

Lim (8x₀ + 4h - 2) = 8x₀ - 2

h → 0

g'(x₀) = 8x - 2

g'(5) = 40 - 2 = 38 ← this is the same result (above)

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### How far is a 5k?

3 ounces short of 1 cup.

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### The sides of a triangle are in the ratio of 1/2:1/3:1/4. If the perimeter is 52 cm, the length or the smallest side is?

I'd first multiply all the fractions by the LCD of 12 to get rid of denominators.

12*½ : 12*⅓ : 12*¼

= 6 : 4 : 3

Now think of the sides as 6k, 4k and 3k respectively. These add up to 52 cm.

6k + 4k + 3k = 52

13k = 52

k = 4

The sides are therefore:

6k → 6(4) = 24 cm

4k → 4(4) = 16 cm

3k → 3(4) = 12 cm

Note: We could have kept the numbers as fractions and you could have still gotten the same answer.

Let the sides be ½x, ⅓x and ¼x respectively.

½x + ⅓x + ¼x = 52

(½ + ⅓ + ¼)x = 52

(6/12 + 4/12 + 3/12)x = 52

(13/12)x = 52

x = 52*(12/13)

x = 52/13 * 12

x = 4*12

x = 48

So the sides are:

½x → 48/2 = 24 cm

⅓x → 48/3 = 16 cm

¼x → 48/4 = 12 cm

I just like working with integers rather than fractions which is why I multiplied it all by 12 at the beginning.

The shortest side is 12 cm

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### How do I solve 15^x-8+5=62 And round 3 places past the decimal?

Exponential and log equations

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• 0

### Solve the system: y=14x2+2x−6 y=−14x2−3x+6?

The parabolas above intersect in two places, at (a , b) and (c , d), where a, b, c, and d are all integers. a + b + c + d =

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### Percentage question ?

Before the increase, the customer spent \$100 for gas (for example). After the increase, it would be \$125. So the customer needs to reduce consumption by \$25.  And 25 is _____ % of 125 ?

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### Two numbers are in the ratio 3:5. If 9 is subtracted from the numbers, the ratio becomes 12:23. The numbers are?

Let the original numbers be 3k and 5k, respectively.

Write an equation for the ratio after subtracting 9 from each number:

(3k-9)/(5k-9) = 12/23

Cross multiply:

23(3k-9) = 12(5k-9)

69k - 207 = 60k - 108

Group like terms:

69k - 60k = 207 - 108

9k = 99

k = 11

3k → 3(11) = 33

5k → 5(11) = 55

The original numbers are 33 and 55.

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### find the sum of (7t + 6 ) + (-4t -2)?

does anyone know?

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### If the side of a square increases by 40%, then the area of the square increases by?

Suppose the length of a square is s. The area of such a square is given by

A =  s²

If s is increased by 40%, the resulting length will be (s + 0.40s)  = 1.40s. Thus, the area of the new square will be

A' = (1.40s)(1.40s) = 1.96s²

Note that we can rewrite as

A' = s² + 0.96s²

This is mathematically equivalent to the original area, s²,  increasing by 96%.

Hope this helps!

• 3

### How many numbers between 100 and 300 begin or end with 2?

It's probably easier to first count the total numbers and then subtract the ones that *don't* start or end with 2.

Let's actually focus on the numbers 100 to 299. That's 200 numbers.

The first digit can be 1 or 2, but only 1 works to not have a starting digit of 2.

1 choice for the first digit.

The second digit can be anything (0 to 9)

10 choices for the second digit.

The third digit can be anything except 2 (0, 1, 3, ..., 9)

9 choices for the third digit.

So multiplying everything out:

1 * 10 * 9 = 90 numbers that don't start or end with 2.

That leaves 110 numbers that *do* start or end with 2.

If you want to enumerate them:

102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 200, 201, 202, ..., 298, 299

110 numbers

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### Solve the equation sin(x+3)=sinx, 0=<x=<2pi to 4 decimal places. The product of the two solutions is:?

Need help with this problem pls

• 3

### graph your equations: 2x + 3y = 7 and y=2x + 27 label them properly, including point of intersection?

2x + 3y = 7 eq1 and y=2x + 27 eq2

plug in eq2 to eq1

2x + 3(2x + 27) = 7

2x + 6x + 81 = 7

8x = 7 - 81

8x = -74