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Mathematics

1 answer:

VARVARA [1.3K]3 years ago

3 0

Problem 5

<h3>Answer: $490</h3>

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Explanation:

When we increase some value by 7%, we are effectively multiplying by 1.07

Note how 100% + 7% = 107% = 1.07

The 1.07 multiplier is then handy to use algebra to find the cost before tax is added

x = cost before tax

1.07x = cost after tax

1.07x = 524.30

x = 524.30/1.07

x = 490

Before tax, the water heater cost $490.

If we apply a 7% tax, then we pay 7% of 490 = 0.07*490 = 34.30 dollars on top of the $490 to get 490+34.30 = 524.30 which matches with the instructions. So our answer is verified.

Or you could use the 1.07 multiplier shortcut to see that 1.07*490 = 524.30

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Problem 6

<h3>Answer: $1301.12</h3>

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Explanation:

His gross pay is $1750. This is the amount he earned before any taxes are applied. The items of Medicare, Social Security, and Federal Withholding Income Tax, are all various taxes that Ian has to pay. So his gross pay will be reduced to some smaller value to get the net pay.

To find the total amount of taxes deducted from his paycheck, we add up the tax percentage values to get: 1.45+6.2+18 = 25.65

This means Ian is taxed at a total rate of 25.65% of his gross pay, and he has to pay 0.2565*1750 = 448.875 = 448.88 dollars to the government. This leaves 1750-448.88 = 1301.12 dollars left over for his net pay.

A slightly alternative way is to note how if Ian pays 25.65% of his gross pay, then he's left with 100% - 25.65% = 74.35% of his paycheck, and 0.7435*1750 = 1,301.125 which would round down to 1301.12

I'm rounding up when it comes to determining the amount of tax to pay, while rounding down any amount Ian has left over for his net pay. This ensures that we consider the worst-case scenario.

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Which graph correctly solves the system of equations below? y = −x2 − 1 y = 2x2 − 4 (6 points)

myrzilka [38]

The equations are

$i) y=-x^2-1\\\\ii) y=2x^2-4$ ,

The graphs of the solutions (x, y) of these equations are 2 parabolas, since the right hand side expressions are polynomials of degree 2.

The solution/s of the system are the x-coordinates of the point/s of intersection of the parabolas.

The solutions of the first equation form a parabola looking downwards (since the coefficient of x^2 is -), and the second, a parabola opening upwards (since the coefficient of x^2 is +).

We can draw both parabolas, but to find the solution we still need to solve the system algebraically.

The algebraic solution of the system is:

$-x^2-1=2x^2-4\\\\3x^2-3=0\\\\3(x^2-1)=0\\\\x^2-1=0$ , so

the solutions are x=-1 and x=1.

The graph of the system is drawn using desmos.com

If we are allowed to use a graphic calculator, we can draw both graphs and point at the solution.