If the base of a rectangle is increased by `10%` and the area is unchanged, then the corresponding altitude must be decreased by -----

Options

A. A)`9 1/11 %`
B. B)`10%`
C. C)`11%`
D. D)`11 1/9 %`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We know that the area of a rectangle is given by the formula:
\[ \text{Area} = \text{Base} \times \text{Altitude} \]
Let’s denote the original base as \( b \) and the original altitude as \( h \). The area can be expressed as:
\[ A = b \times h \]

### Step 2: Increase the Base
According to the problem, the base is increased by \( 10\% \). Therefore, the new base \( b' \) can be calculated as:
\[ b' = b + 0.1b = 1.1b \]

### Step 3: Set the Area Unchanged
Since the area remains unchanged, we can set up the equation for the new area:
\[ A = b' \times h' \]
Where \( h' \) is the new altitude. Since the area is unchanged:
\[ b \times h = 1.1b \times h' \]

### Step 4: Simplify the Equation
We can simplify the equation by dividing both sides by \( b \) (assuming \( b \neq 0 \)):
\[ h = 1.1h' \]

### Step 5: Solve for the New Altitude
Now, we can express \( h' \) in terms of \( h \):
\[ h' = \frac{h}{1.1} \]

### Step 6: Calculate the Decrease in Altitude
To find how much the altitude has decreased, we calculate the difference:
\[ \text{Decrease} = h - h' = h - \frac{h}{1.1} \]
This simplifies to:
\[ \text{Decrease} = h \left(1 - \frac{1}{1.1}\right) \]
\[ = h \left(\frac{1.1 - 1}{1.1}\right) = h \left(\frac{0.1}{1.1}\right) \]

### Step 7: Calculate the Percentage Decrease
To find the percentage decrease, we use the formula:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{h}\right) \times 100\% \]
Substituting the decrease we found:
\[ \text{Percentage Decrease} = \left(\frac{h \left(\frac{0.1}{1.1}\right)}{h}\right) \times 100\% = \left(\frac{0.1}{1.1}\right) \times 100\% \]
\[ = \frac{10}{1.1} \approx 9.09\% \]

### Final Answer
Thus, the corresponding altitude must be decreased by approximately \( 9.09\% \).

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