Math / Algebra

QuestionAn open-top box is to be constructed from a sheet of tin that measures 24 inches by 16 inches by cutting out squares from each corner as shown and then folding up the sides. Let $V(x)$ denote the volume of the resulting box.
Step 2 of 2 : Among the values of $x$ for which $V(x)=0$ , which are physically possible?

Studdy Solution

STEP 1

What is this asking? We're finding the realistic sizes of the squares we can cut from a 24x16 inch sheet to make a box with zero volume. Watch out! Math might give us some crazy answers, but we have to remember we're dealing with a real box!

STEP 2

1. Define the function
2. Find the zeros
3. Check for physical possibility

STEP 3

Alright, so we're building a box!
We start with a sheet of tin that's 24 inches long and 16 inches wide.
We're cutting squares of side length $x$ from each corner.

STEP 4

When we fold up the sides, the length of the box becomes $24 - 2x$ , the width becomes $16 - 2x$ , and the height is just $x$ .

STEP 5

The volume $V(x)$ of a box is length times width times height.
So, our volume function is $V(x) = (24 - 2x) \cdot (16 - 2x) \cdot x$ .

STEP 6

We want to find the values of $x$ where the volume is zero.
That means we're solving $V(x) = 0$ , or $(24 - 2x) \cdot (16 - 2x) \cdot x = 0$ .

STEP 7

This equation is true if any of the factors are zero.
So, we have three possibilities: $24 - 2x = 0$ , $16 - 2x = 0$ , or $x = 0$ .

STEP 8

Let's solve $24 - 2x = 0$ .
Adding $2x$ to both sides gives $24 = 2x$ .
Dividing both sides by 2 gives us $x = 12$ .

STEP 9

Now let's solve $16 - 2x = 0$ .
Adding $2x$ to both sides gives $16 = 2x$ .
Dividing both sides by 2 gives us $x = 8$ .

STEP 10

And of course, we have $x = 0$ .
So our solutions are $x = 0$ , $x = 8$ , and $x = 12$ .

STEP 11

If $x = 0$ , we haven't cut anything out, so we don't have a box.
That makes sense because the volume would be zero!

STEP 12

If $x = 8$ , we're cutting out squares of side length 8 inches from each corner.
The width of the tin sheet is only 16 inches, so cutting out $2 \cdot 8 = 16$ inches would leave us with no width!
So $x = 8$ isn't physically possible.

STEP 13

If $x = 12$ , we're cutting out squares of side length 12 inches.
The length of the tin sheet is 24 inches, so cutting out $2 \cdot 12 = 24$ inches would leave us with no length!
So $x = 12$ isn't physically possible either.

STEP 14

Only $x = 0$ is physically possible, even though it doesn't actually make a box.

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Studdy Solution

STEP 1

What is this asking? We're finding the realistic sizes of the squares we can cut from a 24x16 inch sheet to make a box with zero volume. Watch out! Math might give us some crazy answers, but we have to remember we're dealing with a real box!

STEP 2

1. Define the function
2. Find the zeros
3. Check for physical possibility

STEP 3

Alright, so we're building a box!
We start with a sheet of tin that's 24 inches long and 16 inches wide.
We're cutting squares of side length $x$ from each corner.

STEP 4

When we fold up the sides, the length of the box becomes $24 - 2x$ , the width becomes $16 - 2x$ , and the height is just $x$ .

STEP 5

The volume $V(x)$ of a box is length times width times height.
So, our volume function is $V(x) = (24 - 2x) \cdot (16 - 2x) \cdot x$ .

STEP 6

We want to find the values of $x$ where the volume is zero.
That means we're solving $V(x) = 0$ , or $(24 - 2x) \cdot (16 - 2x) \cdot x = 0$ .

STEP 7

This equation is true if any of the factors are zero.
So, we have three possibilities: $24 - 2x = 0$ , $16 - 2x = 0$ , or $x = 0$ .

STEP 8

Let's solve $24 - 2x = 0$ .
Adding $2x$ to both sides gives $24 = 2x$ .
Dividing both sides by 2 gives us $x = 12$ .

STEP 9

Now let's solve $16 - 2x = 0$ .
Adding $2x$ to both sides gives $16 = 2x$ .
Dividing both sides by 2 gives us $x = 8$ .

STEP 10

And of course, we have $x = 0$ .
So our solutions are $x = 0$ , $x = 8$ , and $x = 12$ .

STEP 11

If $x = 0$ , we haven't cut anything out, so we don't have a box.
That makes sense because the volume would be zero!

STEP 12

If $x = 8$ , we're cutting out squares of side length 8 inches from each corner.
The width of the tin sheet is only 16 inches, so cutting out $2 \cdot 8 = 16$ inches would leave us with no width!
So $x = 8$ isn't physically possible.

STEP 13

If $x = 12$ , we're cutting out squares of side length 12 inches.
The length of the tin sheet is 24 inches, so cutting out $2 \cdot 12 = 24$ inches would leave us with no length!
So $x = 12$ isn't physically possible either.

STEP 14

Only $x = 0$ is physically possible, even though it doesn't actually make a box.

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Studdy Solution

STEP 1

What is this asking? We're finding the *realistic* sizes of the squares we can cut from a 24x16 inch sheet to make a box with *zero* volume. Watch out! Math might give us some *crazy* answers, but we have to remember we're dealing with a *real* box!

STEP 2

1. Define the function2. Find the zeros3. Check for physical possibility

STEP 3

Alright, so we're building a box!We start with a sheet of tin that's **24 inches** long and **16 inches** wide.We're cutting squares of side length xxx from each corner.

STEP 4

When we fold up the sides, the *length* of the box becomes 24−2x24 - 2x24−2x, the *width* becomes 16−2x16 - 2x16−2x, and the *height* is just xxx.

STEP 5

The *volume* V(x)V(x)V(x) of a box is length times width times height.So, our volume function is V(x)=(24−2x)⋅(16−2x)⋅xV(x) = (24 - 2x) \cdot (16 - 2x) \cdot xV(x)=(24−2x)⋅(16−2x)⋅x.

STEP 6

We want to find the values of xxx where the volume is *zero*.That means we're solving V(x)=0V(x) = 0V(x)=0, or (24−2x)⋅(16−2x)⋅x=0(24 - 2x) \cdot (16 - 2x) \cdot x = 0(24−2x)⋅(16−2x)⋅x=0.

STEP 7

This equation is *true* if any of the factors are zero.So, we have three possibilities: 24−2x=024 - 2x = 024−2x=0, 16−2x=016 - 2x = 016−2x=0, or x=0x = 0x=0.

STEP 8

Let's solve 24−2x=024 - 2x = 024−2x=0.Adding 2x2x2x to both sides gives 24=2x24 = 2x24=2x.Dividing both sides by **2** gives us x=12x = 12x=12.

STEP 9

Now let's solve 16−2x=016 - 2x = 016−2x=0.Adding 2x2x2x to both sides gives 16=2x16 = 2x16=2x.Dividing both sides by **2** gives us x=8x = 8x=8.

STEP 10

And of course, we have x=0x = 0x=0.So our solutions are x=0x = 0x=0, x=8x = 8x=8, and x=12x = 12x=12.

STEP 11

If x=0x = 0x=0, we haven't cut anything out, so we don't have a box.That makes sense because the volume would be zero!

STEP 12

If x=8x = 8x=8, we're cutting out squares of side length **8 inches** from each corner.The width of the tin sheet is only **16 inches**, so cutting out 2⋅8=162 \cdot 8 = 162⋅8=16 inches would leave us with *no width*!So x=8x = 8x=8 isn't physically possible.

STEP 13

If x=12x = 12x=12, we're cutting out squares of side length **12 inches**.The length of the tin sheet is **24 inches**, so cutting out 2⋅12=242 \cdot 12 = 242⋅12=24 inches would leave us with *no length*!So x=12x = 12x=12 isn't physically possible either.

STEP 14

Only x=0x = 0x=0 is physically possible, even though it doesn't actually make a box.

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What is this asking? We're finding the realistic sizes of the squares we can cut from a 24x16 inch sheet to make a box with zero volume. Watch out! Math might give us some crazy answers, but we have to remember we're dealing with a real box!

1. Define the function
2. Find the zeros
3. Check for physical possibility

Alright, so we're building a box!
We start with a sheet of tin that's 24 inches long and 16 inches wide.
We're cutting squares of side length $x$ from each corner.

When we fold up the sides, the length of the box becomes $24 - 2x$ , the width becomes $16 - 2x$ , and the height is just $x$ .

The volume $V(x)$ of a box is length times width times height.
So, our volume function is $V(x) = (24 - 2x) \cdot (16 - 2x) \cdot x$ .

We want to find the values of $x$ where the volume is zero.
That means we're solving $V(x) = 0$ , or $(24 - 2x) \cdot (16 - 2x) \cdot x = 0$ .

This equation is true if any of the factors are zero.
So, we have three possibilities: $24 - 2x = 0$ , $16 - 2x = 0$ , or $x = 0$ .

Let's solve $24 - 2x = 0$ .
Adding $2x$ to both sides gives $24 = 2x$ .
Dividing both sides by 2 gives us $x = 12$ .

Now let's solve $16 - 2x = 0$ .
Adding $2x$ to both sides gives $16 = 2x$ .
Dividing both sides by 2 gives us $x = 8$ .

And of course, we have $x = 0$ .
So our solutions are $x = 0$ , $x = 8$ , and $x = 12$ .

If $x = 0$ , we haven't cut anything out, so we don't have a box.
That makes sense because the volume would be zero!

If $x = 8$ , we're cutting out squares of side length 8 inches from each corner.
The width of the tin sheet is only 16 inches, so cutting out $2 \cdot 8 = 16$ inches would leave us with no width!
So $x = 8$ isn't physically possible.

If $x = 12$ , we're cutting out squares of side length 12 inches.
The length of the tin sheet is 24 inches, so cutting out $2 \cdot 12 = 24$ inches would leave us with no length!
So $x = 12$ isn't physically possible either.

Only $x = 0$ is physically possible, even though it doesn't actually make a box.