Question

1. find the coordinates of b if m is the midpoint of ab.

a (7, 5) and m (9, 7)
find the slope and midpoint of the following points: r (-3, 5) and s (1, 9)

1. slope
2. midpoint
3. gerry wants to calculate the volume of air in a building, shown below,

so that he can decide on the size of new furnace.
the building has a height of 12 feet, length of 40 feet, and width of 75
feet. in cubic feet, what is the volume of the building?
volume of the bottom (rectangular prism) ________
volume of the top (1/2 cylinder) ________
total volume ________

Explanation:

16. Find the coordinates of \( B \) if \( M \) is the mid - point of \( AB \)

Let \( A=(x_1,y_1)=(7,5) \) and \( M=(x_m,y_m)=(9,7) \). If \( M \) is the mid - point of \( AB \) and \( B=(x_2,y_2) \), the mid - point formula is \( x_m=\frac{x_1 + x_2}{2} \) and \( y_m=\frac{y_1 + y_2}{2} \).

• For the \( x \) - coordinate:
• Using \( x_m=\frac{x_1 + x_2}{2} \), substitute \( x_m = 9\) and \(x_1 = 7\).
• \(9=\frac{7 + x_2}{2}\).
• Multiply both sides by \(2\): \(18=7 + x_2\).
• Subtract \(7\) from both sides: \(x_2=18 - 7=11\).
• For the \( y \) - coordinate:
• Using \( y_m=\frac{y_1 + y_2}{2}\), substitute \( y_m = 7\) and \(y_1 = 5\).
• \(7=\frac{5 + y_2}{2}\).
• Multiply both sides by \(2\): \(14=5 + y_2\).
• Subtract \(5\) from both sides: \(y_2=14 - 5 = 9\).

17. Slope of the line passing through \( R(-3,5)\) and \( S(1,9)\)

The slope formula is \(m=\frac{y_2-y_1}{x_2 - x_1}\). Let \(R=(x_1,y_1)=(-3,5)\) and \(S=(x_2,y_2)=(1,9)\).

• Substitute into the formula: \(m=\frac{9 - 5}{1-(-3)}\).
• Simplify the numerator and denominator: \(m=\frac{4}{1 + 3}=\frac{4}{4}=1\).

18. Mid - point of the line segment \( RS\)

Using the mid - point formula \(M=(\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})\). Let \(R=(x_1,y_1)=(-3,5)\) and \(S=(x_2,y_2)=(1,9)\).

• For the \(x\) - coordinate: \(\frac{-3+1}{2}=\frac{-2}{2}=-1\).
• For the \(y\) - coordinate: \(\frac{5 + 9}{2}=\frac{14}{2}=7\). So the mid - point is \((-1,7)\).

19. Volume of the building

• Volume of the bottom (rectangular prism):

The formula for the volume of a rectangular prism is \(V=l\times w\times h\). Here, \(l = 40\) ft, \(w = 75\) ft, \(h = 12\) ft.
\(V_{prism}=40\times75\times12=(40\times75)\times12 = 3000\times12=36000\) cubic feet.

• Volume of the top (\(\frac{1}{2}\) cylinder):

The formula for the volume of a cylinder is \(V=\pi r^2h\). The diameter of the semi - circular part is equal to the length of the rectangular base, so \(d = 40\) ft, \(r=\frac{d}{2}=20\) ft, and \(h = 75\) ft (the width of the rectangular prism).
\(V_{semi - cylinder}=\frac{1}{2}\pi r^2h=\frac{1}{2}\times\pi\times(20)^2\times75=\frac{1}{2}\times\pi\times400\times75=15000\pi\approx15000\times3.14 = 47100\) cubic feet.

• Total Volume:

\(V_{total}=V_{prism}+V_{semi - cylinder}=36000 + 47100=83100\) cubic feet.

Answer:

1. \(B=(11,9)\)
2. \(1\)
3. \((-1,7)\)

19.

• Volume of the bottom (rectangular prism): \(36000\) cubic feet
• Volume of the top (\(\frac{1}{2}\) cylinder): \(47100\) cubic feet
• Total Volume: \(83100\) cubic feet