QuestionAn open-top box is to be constructed from a sheet of tin that measures 42 inches by 24 inches by cutting out squares from each corner as shown and then folding sides. Let denote the volume of the resulting box. inches
Studdy Solution
STEP 1
What is this asking? We need to find the size of the squares to cut out from a tin sheet to make an open-top box with the *biggest* possible volume! Watch out! Don't forget that cutting out squares changes all the dimensions of the box!
STEP 2
1. Set up the volume formula
2. Optimize the volume
STEP 3
Alright, let's **define** our variables! is the **length** of the tin sheet, which is inches, and is the **width**, which is inches.
We're cutting out squares of side length from each corner.
STEP 4
Now, think about what happens when we fold the tin sheet.
The **length** of the box becomes , because we're cutting out two squares from each side of the length.
Similarly, the **width** of the box becomes .
The **height** of the box is just , the size of the squares we cut out.
STEP 5
The **volume** of a box is length times width times height.
So, our **volume formula** is:
Let's plug in our **length** and **width**:
STEP 6
To find the *biggest* volume, we need to find where our volume function hits its **maximum value**.
Let's **expand** the formula to make it easier to work with:
\begin{align*} V(x) &= (42 - 2x)(24 - 2x)x \\ &= (1008 - 84x - 48x + 4x^2)x \\ &= 4x^3 - 132x^2 + 1008x\end{align*}
STEP 7
Now, we'll take the **derivative** of with respect to .
The derivative tells us the **slope** of the function, and we're looking for where the slope is **zero**, which indicates a potential maximum or minimum:
STEP 8
To find where , we can use the **quadratic formula**:
In our case, , , and .
Plugging these values in, we get:
\begin{align*} x &= \frac{264 \pm \sqrt{(-264)^2 - 4 \cdot 12 \cdot 1008}}{2 \cdot 12} \\ &= \frac{264 \pm \sqrt{69696 - 48384}}{24} \\ &= \frac{264 \pm \sqrt{21312}}{24} \\ &= \frac{264 \pm 146}{24}\end{align*}
This gives us two possible values for : and .
STEP 9
Since the width of the tin sheet is inches, cutting out squares of side length inches doesn't make sense!
So, the size of the squares we need to cut out is inches.
STEP 10
The size of the squares to cut out to maximize the volume of the box is approximately **4.92 inches**.
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