Please solve for t this is very important

butalik [34]

When we say two quantities $x$ and $y$ are proportional to one another, there are two ways we could mean this.

- If $x$ is *directly* proportional to $y$ , then we mean that as $x$ changes, $y$ changes in the same direction. In other words, if $x$ is in/decreased, then $y$ also in/decreases. The rate of in/decrease doesn't have to be one-to-one; for example, we could have $x$ increase by 1 unit while $y$ would proportionally increase by 5 units. In this case, we'd have $y=5x$ .

- On the other hand, if $x$ is *inversely* proportional to $y$ , then a change in $x$ results in a change in $y$ that goes in the opposite direction. A common example involves taking a rectangle of constant area and adjusting the width $x$ and length $y$ . If the rectangle has an area of 5 square units, then we could have $x=5$ and $y=1$ , or we could have $x=4$ and $y=\dfrac54$ , or $x=3$ and $y=\dfrac53$ , or any combination of $x$ and $y$ such that $xy=5$ is satisfied.

In both cases, we call 5 the "constant of proportionality".

On to your exercises:

(1a) Looks like $a$ stands for number of apples. Then we're told that the cost $c=2.3a$ , which means that for every apple, the cost increases by 2.3 dollars. So the constant of proportionality is 2.3. In the language of proportionality, we could then say that the cost of apples is directly proportional to the number of apples by a factor of 2.3.

(1b) 5.4

(1c) 12.5

6 0

3 years ago

What is 1/10 of 1000 what is 1% out of 100

Ulleksa [173]

To get 1/10 -> 1000/10 =100
and 1% of 100 -> 100/100 = 1 or 1%

100 for question 1
1 for q2

3 0

3 years ago

Leonard did this work to solve an equation. Did he make an error?

nalin [4]

Yes, Leonard did make an error
It should look like this:
-20x + 35 + 8 = 3x
Hope this helps!

8 0

3 years ago

Find all pairs of real numbers (a,b) such that (x-a)^2+(2x-b)^2=(x-3)^2+(2x)^2

Mrrafil [7]

Answer:

$(3,0)$

$(\frac{-9}{5},\frac{12}{5})$

Step-by-step explanation:

Let's expand both sides.

I'm going to use the following identity to expand the binomial squared expressions: $(u+v)^2=u^2+2uv+v^2$ or $(u-v)^2=u^2-2uv+v^2$ .

Left-hand side:

$(x-a)^2+(2x-b)^2$

$(x^2-2ax+a^2)+((2x)^2-2b(2x)+b^2)$

$x^2-2ax+a^2+4x^2-4bx+b^2$

Reorder so $x^2$ 's are together and that $x[tex]'s are together.[tex](x^2+4x^2)+(-2ax-4bx)+(a^2+b^2)$

$5x^2+(-2a-4b)x+(a^2+b^2)$

Right-hand side:

$(x-3)^2+(2x)^2$

$x^2-2(3)x+9+4x^2$

$x^2-6x+9+4x^2$

Reorder so $x^2$ 's are together and that $x[tex]'s are together.[tex](x^2+4x^2)+(-6x)+9$

$5x^2-6x+9$

Now let's compare both sides.

If we want both sides to appear exactly the same we need to choose values $a$ and $b$ such the following are true equations:

$-2a-4b=-6$

$a^2+b^2=9$

So if we solve the system we can find the values $a$ and $b$ such that the left=right.

Let's solve the first equation for $a$ in terms of $b$ .

Add $2a$ on both sides:

$-4b=-6+2a$

Divide both sides by -4:

$b=\frac{-6+2a}{-4}$

Reduce (divide top and bottom by -2):

$b=\frac{3-a}{2}$

Now let's plug this into second equation:

$a^2+b^2=9$

$a^2+(\frac{3-a}{2})^2=9$

$a^2+\frac{9-6a+a^2}{4}=9$ (I used the identity $(u-v)^2=u^2-2uv+v^2$ )

Multiply both sides by 4 to clear the fractions from the problem:

$4a^2+(9-6a+a^2)=36$

Combine like terms on left hand side:

$4a^2+a^2-6a+9=36$

$5a^2-6a+9=36$

Subtract 36 on both sides:

$5a^2-6a-27=0$

Now let's try to factor.

We are going to try to find two numbers that multiply to be 5(-27) and add to be -6.

5(-27)=(5*3)(-9)=15(-9)=-15(9) while -15+9=-6.

So let's replace $-6a$ with $-15a+9a$ and factor by grouping.

$5a^2-15a+9a-27=0$

$5a(a-3)+9(a-3)=0$

$(a-3)(5a+9)=0$

This implies $a-3=0$ or $5a+9=0$ .

Solving the first is easy. Just ad 3 on both sides to get: $a=3$ .

The second requires two steps. Subtract 9 and then divide by 5 on both sides.

$5a=-9$

$a=\frac{-9}{5}$ .

So let's go back to finding $b$ now that we know the $a$ values.

If $a=3$ and $b=\frac{3-a}{2}$ ,

then $b=\frac{3-3}{2}=0$ .

So one ordered pair $(a,b)$ that satisfies the equation is:

$(3,0)$ .

If $a=\frac{-9}{5}$ and $b=\frac{3-a}{2}$ ,

then $b=\frac{3-\frac{-9}{5}}{2}$ .

Let's multiply top and bottom by 5 to clear the mini-fraction.

$b=\frac{15-(-9)}{10}$

$b=\frac{24}{10}$

$b=\frac{12}{5}$

So one ordered pair $(a,b)$ that satisfies the equation is:

$(\frac{-9}{5},\frac{12}{5})$ .

5 0

4 years ago

Can somebody help

mihalych1998 [28]

Answer:

-0.75

Step-by-step explanation:

The table values listed for the left side expression and the right side expression are closest together when the value of x is -0.75.

__

A graphing calculator shows the expressions both have a value near -0.658 when x ≈ - 0.775. The closest table value to x = -0.775 is x = -0.75.

4 0

3 years ago

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