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3 years ago

Solve z^2-2z-35=0 using zero product

Mathematics

1 answer:

Stels [109]3 years ago

3 0

Answer: z = 7 or z=-5

Step-by-step explanation:

First you need to factor the polynomial.

To factor $z^{2}$ - 2z -35 You will need to find two numbers that their product is -35 and their sum is -2 and the numbers -7 and 5 works out.

Rewrite the polynomial as,

$z^{2}$ - 7z + 5z - 35 = 0 factor the left by grouping

( $z^{2}$ -7z) (5z -35) = 0

z(z -7) 5(z - 7) = 0 Factor out z-7

(z-7)(z+5) = 0 Now apply the zero product by setting each of them equal 0.

z-7= 0 or z+5=0

+7 +7 -5 -5

z = 7 or z= -5

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Two different radioactive isotopes decay to 10% of their respective original amounts. Isotope A does this in 33 days, while isot

Andrews [41]

Answer:

The approximate difference in the half-lives of the isotopes is 66 days.

Step-by-step explanation:

The decay of an isotope is represented by the following differential equation:

$\frac{dm}{dt} = -\frac{t}{\tau}$

Where:

$m$ - Current mass of the isotope, measured in kilograms.

$t$ - Time, measured in days.

$\tau$ - Time constant, measured in days.

The solution of the differential equation is:

$m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }$

Where $m_{o}$ is the initial mass of the isotope, measure in kilograms.

Now, the time constant is cleared:

$\ln \frac{m(t)}{m_{o}} = -\frac{t}{\tau}$

$\tau = -\frac{t}{\ln \frac{m(t)}{m_{o}} }$

The half-life of a isotope ( $t_{1/2}$ ) as a function of time constant is:

$t_{1/2} = \tau \cdot \ln2$

$t_{1/2} = -\left(\frac{t}{\ln\frac{m(t)}{m_{o}} }\right) \cdot \ln 2$

The half-life difference between isotope B and isotope A is:

$\Delta t_{1/2} = \left| -\left(\frac{t_{A}}{\ln \frac{m_{A}(t)}{m_{o,A}} } \right)\cdot \ln 2+\left(\frac{t_{B}}{\ln \frac{m_{B}(t)}{m_{o,B}} } \right)\cdot \ln 2\right|$

If $\frac{m_{A}(t)}{m_{o,A}} = \frac{m_{B}(t)}{m_{o,B}} = 0.9$ , $t_{A} = 33\,days$ and $t_{B} = 43\,days$ , the difference in the half-lives of the isotopes is:

$\Delta t_{1/2} = \left|-\left(\frac{33\,days}{\ln 0.90} \right)\cdot \ln 2 + \left(\frac{43\,days}{\ln 0.90} \right)\cdot \ln 2\right|$

$\Delta t_{1/2} \approx 65.788\,days$

The approximate difference in the half-lives of the isotopes is 66 days.

4 0

3 years ago

Read 2 more answers

The radius of a circle with area A can be approximated using the formula r = the square root of A/3 . Estimate the radius of a w

Arturiano [62]

Answer: 12

Step-by-step explanation:

Given:

A = 452

r = √A/3

Step 1: Substitute 452sqf for A

r = √452/3

Step 2: Solve

452 ÷ 3 = 150.66∞

√150.66∞ = 12.27∞

12.27∞ rounded to the nearest integer equal 12

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3 years ago

How can Diego package all the 64 cookies so that each bag has the same number of them? How many bags can we make and how many co

anygoal [31]

Answer:

see below

Step-by-step explanation:

There are 7 divisors of 64. Each represents a possible number of bags of cookies:

1 bag of 64 cookies
2 bags of 32 cookies
4 bags of 16 cookies
8 bags of 8 cookies
16 bags of 4 cookies
32 bags of 2 cookies
64 bags of 1 cookie

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3 years ago

Find the area of this shape.

aniked [119]

Answer:

Step-by-step explanation:

Bottom triangle

Area = b * height

b = 4 + 4 = 8 cm

h = 5.75

Area = 1/2 * 5.75 * 8

Area = 23 cm^2

Trapezoid

b1 = 8

b2 = 4

h = 2

Area = (b1 + b2)*h/ 2

Area = (8 + 4)*2 /2

Area = 12

Total area = 23 + 12

Total area = 35

7 0

3 years ago

What is an equation of the line that passes through the point (2,−6) and is parallel to the line x-2y=8

lana [24]

Answer:

$y=\frac{1}{2}x-7$ , or x-2y=14

Step-by-step explanation:

Hi there!

We want to find the equation of the line that passes through the point (2, -6) and is parallel to the line x-2y=8

First, we need to find the slope of x-2y=8, since parallel lines have the same slopes

We can convert the equation from standard form (ax+by=c) to slope-intercept form (y=mx+b, where m is the slope and b is the y intercept), in order to help us find the slope of the line

Start by subtracting x from both sides

-2y=-x+8

Divide both sides by -2

y= $\frac{1}{2}x$ -4

The slope of the line x-2y=8 is 1/2

It's also the slope of the line parallel to it.

Since we know the slope of the line, we can plug it into the equation for slope intercept form:

$y=\frac{1}{2}x+b$

Now we need to find b.

As the equation of the line passes through (2, -6), we can use it to help solve for b

Substitute -6 as y and 2 as x:

$-6=\frac{1}{2}(2)+b$

Multiply

-6=1+b

Subtract 1 from both sides

-7=b

Substitute -7 as b into the equation:

$y=\frac{1}{2}x-7$

The equation can be left as that, or you can convert it into standard form if you wish.

In that case, you will need to move 1/2x to the other side:

$-\frac{1}{2}x+y=-7$

A rule about the coefficients a, b, and c in standard form is that a (coefficient in front of x) CANNOT be negative, and every coefficient must be an integer (a whole number, not a fraction or decimal).

So multiply both sides by -2 in order to clear the fraction, as well as change the sign of a

x-2y=14

Hope this helps!

8 0

3 years ago