Question
The function a(b) relates the area of a trapezoid with a given height of 10 and one base length of 7 with the length of its other base.
It takes as input the other base value, and returns as output the area of the trapezoid.
а (6) = 10.0+7
Which equation below represents the inverse function b(a), which takes the trapezoid's area as input and returns as output the length of the other base?
06(a) = a-5
06(a) = 9+5
06(a) =½+7
06(a) =9-7
It takes as input the other base value, and returns as output the area of the trapezoid.
а (6) = 10.0+7
Which equation below represents the inverse function b(a), which takes the trapezoid's area as input and returns as output the length of the other base?
06(a) = a-5
06(a) = 9+5
06(a) =½+7
06(a) =9-7
Answers
GPT-4o mini
To find the inverse function \( b(a) \), where the area \( A \) of a trapezoid is related to its bases and height, we start with the area formula for a trapezoid:
\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]
For your specific trapezoid, we know:
- \( h = 10 \)
- \( b_1 = 7 \)
- \( b_2 \) is the length of the other base, which is what we will vary.
Plugging in the known values, the area becomes:
\[
A = \frac{1}{2} \times (7 + b_2) \times 10
\]
Simplifying this, we have:
\[
A = 5 \times (7 + b_2) = 35 + 5b_2
\]
Now we solve for \( b_2 \) in terms of \( A \):
\[
A = 35 + 5b_2
\]
Subtract \( 35 \) from both sides:
\[
A - 35 = 5b_2
\]
Now divide by \( 5 \):
\[
b_2 = \frac{A - 35}{5}
\]
This function, \( b(a) = \frac{A - 35}{5} \), represents the inverse, where \( A \) is the area and \( b_2 \) is the length of the other base.
Now, we can restate it as:
\[
b(a) = \frac{a - 35}{5}
\]
Looking at the provided options:
1. \( 06(a) = a - 5 \)
2. \( 06(a) = 9 + 5 \)
3. \( 06(a) = \frac{1}{2} + 7 \)
4. \( 06(a) = 9 - 7 \)
None of these match our derived function. However, let's denote \( 06(a) \) as \( b(a) \):
To find a relationship, we have derived \( b(a) = \frac{a - 35}{5} \). After examining if any provided functions can be a representation, we learn that they do not actually translate into this derived function.
Hence, none of the options accurately represent the inverse function derived from the area. If there may be some other forms or numerical values that were not conveyed, those could potentially form matching connections.
Would you need further assistance in deducing or confirming another part of the process?
\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]
For your specific trapezoid, we know:
- \( h = 10 \)
- \( b_1 = 7 \)
- \( b_2 \) is the length of the other base, which is what we will vary.
Plugging in the known values, the area becomes:
\[
A = \frac{1}{2} \times (7 + b_2) \times 10
\]
Simplifying this, we have:
\[
A = 5 \times (7 + b_2) = 35 + 5b_2
\]
Now we solve for \( b_2 \) in terms of \( A \):
\[
A = 35 + 5b_2
\]
Subtract \( 35 \) from both sides:
\[
A - 35 = 5b_2
\]
Now divide by \( 5 \):
\[
b_2 = \frac{A - 35}{5}
\]
This function, \( b(a) = \frac{A - 35}{5} \), represents the inverse, where \( A \) is the area and \( b_2 \) is the length of the other base.
Now, we can restate it as:
\[
b(a) = \frac{a - 35}{5}
\]
Looking at the provided options:
1. \( 06(a) = a - 5 \)
2. \( 06(a) = 9 + 5 \)
3. \( 06(a) = \frac{1}{2} + 7 \)
4. \( 06(a) = 9 - 7 \)
None of these match our derived function. However, let's denote \( 06(a) \) as \( b(a) \):
To find a relationship, we have derived \( b(a) = \frac{a - 35}{5} \). After examining if any provided functions can be a representation, we learn that they do not actually translate into this derived function.
Hence, none of the options accurately represent the inverse function derived from the area. If there may be some other forms or numerical values that were not conveyed, those could potentially form matching connections.
Would you need further assistance in deducing or confirming another part of the process?