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emmainna [20.7K]

3 years ago

7

Tommy has three times as many video games as Griffin. Fletcher has eight less than Tommy. The total number of video games is les

s than 40. Circle the number that represents the greatest number of video games that Fletcher can own?

1 answer:

zimovet [89]3 years ago

6 0

Answer:

x = 6.86

Number of video games Fletcher has = 12.58

Step-by-step explanation:

Let

Number of video games Griffin has = x

Number of video games Tommy has = 3x

Number of video games Fletcher has = 3x - 8

Total number of video games is > 40

Griffin + Tommy + Fletcher > 40

x + 3x + (3x - 8) > 40

4x + 3x - 8 > 40

7x > 40 + 8

7x > 48

x > 48 / 7

x > 6.86

Number of video games Fletcher has = 3x - 8

= 3(6.86) - 8

= 20.58 - 8

= 12.58

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Answer:

When r=0.2, m=0.016g

When m=0.25, r=0.5cm

When r=0.7, m=0.686g

When m=11.664, r=1.8cm

Step-by-step explanation:

General outline of steps for proportionality problems:

Identify the type or proportionality.
Find the proportionality constant using a known input/output pair.
Use the proportionality equation to find other unknowns.

<h3><u>Background on proportionality relationships</u></h3>

There are two main types of proportionality, "direct" and "inverse", and then there are modifications that can be made to them. Several examples are listed below:

<u>Direct proportionality examples</u>

y is directly proportional to x: $y=kx$
y is directly proportional to the square of x: $y=kx^2$
y is directly proportional to the cube of x: $y=kx^3$

<u>Inverse proportionality examples</u>

y is inversely proportional to x: $y=\dfrac{k}{x}$
y is inversely proportional to the square of x: $y=\dfrac{k}{x^2}$

In each case, irregardless of which type, the two quantities are related with some extra letter "k", called the proportionality constant. Either way, the proportionality constant "k" is always in the numerator, and the quantity is either multiplied to or divided from the proportionality constant "k".

Notice that for direct proportionality, in each case, the equation always ends up as "k times" the quantity.

On the other hand, notice that for inverse proportionality, in each case, the equation always ends up as "k divided by" the quantity.

<h3><u>Step 1. Identify the type or proportionality </u><u>(Setting up our proportionality equation)</u></h3>

The problem says "... the mass, <em>m</em> g, of a sphere is directly proportional to the cube of its radius, <em>r</em> cm...", so our equation will look like $m=kr^3$ .

<h3><u>Step 2. Finding the proportionality constant</u></h3>

To find the proportionality constant for our situation, one must know a full input/output pair. Notice that in the 4th column, $r=1.5$ and $m=6.75$ .

Substituting these values into our equation, we can find "k".

$(6.75)=k(1.5)^3$

$6.75=k*3.375$

$\dfrac{6.75}{3.375}= \dfrac{k*3.375}{3.375}$

$2=k$

So, the proportionality constant for this situation is 2, and our equation for this situation becomes: $m=2r^3$

<h3><u>Step 3. Finding the other inputs/outputs</u></h3>

Now that we know the proportionality constant for this situation, if we have either the input OR the output, we can solve for the other unknown.

<u>r=0.2</u>

$m=2r^3$

$m=2(0.2)^3$

$m=2(0.008)$

$m=0.016$

Recall the the question said that the mass, m, was measured in grams, and the radius, r, was measured in centimeters. So, if the radius is 0.2cm, then the mass of the sphere would be 0.016g.

<u>m=0.25</u>

$m=2r^3$

$(0.25)=2r^3$

$\dfrac{0.25}{2}=\dfrac{2r^3}{2}$

$0.125=r^3$

$\sqrt[3]{0.125} = \sqrt[3]{r^3}$

$0.5=r$

So, if the mass of the sphere were 0.25g, the radius of the sphere would be 0.5cm.

<u>r=0.7</u>

$m=2r^3$

$m=2(0.7)^3$

$m=2(0.343)$

$m=0.686$

So, if the radius is 0.7cm, then the mass of the sphere would be 0.686g.

<u>m=11.664</u>

$m=2r^3$

$(11.664)=2r^3$

$\dfrac{11.664}{2}=\dfrac{2r^3}{2}$

$5.832=r^3$

$\sqrt[3]{5.832} = \sqrt[3]{r^3}$

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So, if the mass of the sphere were 11.664g, the radius of the sphere would be 1.8cm.

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Answer:

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Step-by-step explanation:

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3 years ago