Question

what is the missing length?
6.4 mi
10 mi
area = 107 mi²
z = \boxed{} miles

Explanation:

Step1: Recall trapezoid area formula

The area \( A \) of a trapezoid is given by \( A=\frac{(a + b)h}{2} \), where \( a \) and \( b \) are the lengths of the two parallel sides, and \( h \) is the height. Here, one parallel side \( a = 6.4 \) mi, height \( h = 10 \) mi, area \( A = 107 \) \( \text{mi}^2 \), and the other parallel side (the bottom base) can be expressed as \( 6.4 + z \) (since \( z \) is the extra length on the right side of the top base's projection). Wait, actually, let's correct: let the two bases be \( b_1 = 6.4 \) and \( b_2=6.4 + z \)? No, wait, looking at the diagram, the top base is \( 6.4 \) mi, the height is \( 10 \) mi, and the bottom base is \( 6.4 + z \)? Wait, no, maybe the bottom base is \( 6.4 + z \)? Wait, no, actually, the bottom base is composed of the top base's length plus \( z \) (since the dashed line is the height, so the horizontal segment from the end of the top base to the projection of the height is \( z \)? Wait, maybe better to let the two bases be \( b_1 = 6.4 \) and \( b_2=6.4 + z \)? Wait, no, let's re - express the formula. Let the two parallel sides (bases) be \( a \) and \( b \), height \( h \). So \( A=\frac{(a + b)h}{2} \). We know \( A = 107 \), \( h = 10 \), \( a = 6.4 \), and \( b=6.4 + z \)? No, wait, maybe the bottom base is \( 6.4+z \)? Wait, no, looking at the diagram, the top base is \( 6.4 \) mi, the height is \( 10 \) mi, and the bottom base is \( 6.4 + z \)? Wait, no, actually, the bottom base is \( 6.4 + z \) (the top base is \( 6.4 \), and the bottom base extends \( z \) beyond the right - hand side of the top base's vertical projection). So substituting into the area formula:

\( 107=\frac{(6.4+(6.4 + z))\times10}{2} \)

Wait, no, that's wrong. Wait, maybe the bottom base is \( 6.4+z \), but actually, the two bases are \( 6.4 \) and \( (6.4 + z) \)? No, that can't be. Wait, maybe the bottom base is \( 6.4+z \), but let's do it step by step.

First, multiply both sides of the area formula by 2:

\( 2A=(a + b)h \)

Substitute \( A = 107 \), \( h = 10 \):

\( 2\times107=(6.4 + b)\times10 \)

Step2: Solve for the sum of the bases

Calculate the left - hand side: \( 2\times107 = 214 \)

So \( 214=(6.4 + b)\times10 \)

Divide both sides by 10:

\( \frac{214}{10}=6.4 + b \)

\( 21.4=6.4 + b \)

Then, solve for \( b \) (the length of the bottom base):

\( b=21.4 - 6.4=15 \)

Step3: Find the value of \( z \)

Since the bottom base \( b = 6.4+z \) (from the diagram, the top base is \( 6.4 \), and the bottom base is the top base plus \( z \) on the right - hand side), we have:

\( 15=6.4 + z \)

Subtract \( 6.4 \) from both sides:

\( z=15 - 6.4 = 8.6 \)

Wait, let's check the formula again. Maybe the two bases are \( 6.4 \) and \( (6.4 + z) \)? No, wait, maybe the bottom base is \( 6.4+z \), but actually, the correct way is: Let the two bases be \( b_1 = 6.4 \) and \( b_2=6.4 + z \), height \( h = 10 \). Then area \( A=\frac{(b_1 + b_2)h}{2} \)

Substitute \( A = 107 \), \( h = 10 \), \( b_1 = 6.4 \):

\( 107=\frac{(6.4+(6.4 + z))\times10}{2} \)

Simplify the right - hand side:

\( \frac{(12.8 + z)\times10}{2}=5\times(12.8 + z)=64+5z \)

So we have the equation:

\( 107 = 64+5z \)

Subtract 64 from both sides:

\( 107 - 64=5z \)

\( 43 = 5z \)

Wait, that's a mistake! Oh no, I messed up the base lengths. Let's start over.

The correct formula for the area of a trapezoid is \( A=\frac{(b_1 + b_2)h}{2} \), where \( b_1 \) and \( b_2 \) are the two parallel sides (bases). In the diagram, the top base \( b_1 = 6.4 \) mi, the height \( h = 10 \) mi, and the bottom base \( b_2=6.4 + z \)? No, that's not right. Wait, looking at the diagram, the dashed line is the height (10 mi), and the bottom base is composed of the top base (6.4 mi) plus \( z \) (the horizontal segment from the end of the top base to the end of the bottom base on the right). So the two bases are \( b_1 = 6.4 \) and \( b_2=6.4 + z \)? No, that would mean the bottom base is longer than the top base by \( z \) on both sides? No, the diagram shows that the height is drawn from the left end of the top base? No, the right angle is at the bottom, so the height is vertical, and the top base is horizontal with length 6.4 mi. The bottom base is horizontal, and the segment from the projection of the height (the red right angle) to the right end of the bottom base is \( z \). So the bottom base length is \( 6.4+z \) (since the left part of the bottom base is equal to the top base, and the right part is \( z \))? Wait, no, that's incorrect. Actually, the two bases are \( b_1 = 6.4 \) and \( b_2=6.4 + z \), and the height is 10. But when we plug into the area formula:

\( A=\frac{(b_1 + b_2)h}{2}=\frac{(6.4+(6.4 + z))\times10}{2}=\frac{(12.8 + z)\times10}{2}=5\times(12.8 + z)=64 + 5z \)

We know \( A = 107 \), so:

\( 64+5z=107 \)

Subtract 64 from both sides:

\( 5z=107 - 64=43 \)

Then \( z=\frac{43}{5}=8.6 \)? Wait, no, \( 107-64 = 43 \), \( 43\div5 = 8.6 \). Wait, but let's check with the correct base lengths. Wait, maybe the bottom base is \( 6.4 + z \), but actually, the two bases are \( 6.4 \) and \( (6.4 + z) \), and the height is 10. Let's verify the area with \( z = 8.6 \):

\( b_2=6.4 + 8.6=15 \)

\( A=\frac{(6.4 + 15)\times10}{2}=\frac{21.4\times10}{2}=107 \), which matches the given area. So that's correct.

Step1: Write the trapezoid area formula

The area \( A \) of a trapezoid is given by \( A=\frac{(b_1 + b_2)h}{2} \), where \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides (bases) and \( h \) is the height.

Step2: Substitute the known values

We know \( A = 107\space\text{mi}^2 \), \( h = 10\space\text{mi} \), \( b_1 = 6.4\space\text{mi} \), and \( b_2=6.4 + z \) (since the bottom base is the top base length plus \( z \)). Substitute into the formula:

\( 107=\frac{(6.4+(6.4 + z))\times10}{2} \)

Step3: Simplify the right - hand side

First, simplify the numerator inside the fraction: \( 6.4+(6.4 + z)=12.8 + z \)

Then the formula becomes: \( 107=\frac{(12.8 + z)\times10}{2} \)

Simplify \( \frac{(12.8 + z)\times10}{2} \) to \( 5\times(12.8 + z)=64+5z \)

So we have the equation: \( 107 = 64+5z \)

Step4: Solve for \( z \)

Subtract 64 from both sides of the equation:

\( 107 - 64=5z \)

\( 43 = 5z \)

Divide both sides by 5:

\( z=\frac{43}{5}=8.6 \)

Answer:

\( 8.6 \)