Question

rectangular pyramids x and y have equal volumes.
pyramid x: base length 13 cm, base width 12 cm, height 20 cm
pyramid y: base length 12 cm, base width 10 cm, height h
which is the height, h, of pyramid y?
a. 17 cm
b. 20 cm
c. 23 cm
d. 26 cm

Explanation:

Step1: Recall Volume of Pyramid

The volume \( V \) of a rectangular pyramid is given by \( V=\frac{1}{3} \times \text{base area} \times \text{height} \). For Pyramid X, base length \( l = 13\) cm, base width \( w = 12\) cm, height \( H = 20\) cm. So base area of X is \( l\times w=13\times12 \), and volume \( V_X=\frac{1}{3}\times13\times12\times20 \).

Step2: Calculate Volume of X

Compute \( V_X \): \( \frac{1}{3}\times13\times12\times20 = 13\times4\times20=13\times80 = 1040 \) \( \text{cm}^3 \).

Step3: Volume of Y

For Pyramid Y, base length \( l' = 12\) cm, base width \( w' = 10\) cm, height \( h \) (unknown). Base area is \( l'\times w' = 12\times10 \), volume \( V_Y=\frac{1}{3}\times12\times10\times h \).

Step4: Set Volumes Equal

Since \( V_X = V_Y \), we have \( \frac{1}{3}\times12\times10\times h=1040 \). Simplify left side: \( 4\times10\times h = 40h \). So \( 40h = 1040 \).

Step5: Solve for h

Divide both sides by 40: \( h=\frac{1040}{40}=26 \)? Wait, no, wait. Wait, Pyramid X: wait, maybe I made a mistake. Wait, Pyramid X: base is 13 and 12, height 20. Pyramid Y: base 12 and 10, height h. Wait, no, wait the volume formula: \( V=\frac{1}{3}Bh \), where \( B \) is base area. So for X: \( B_X = 13\times12 = 156 \), \( V_X=\frac{1}{3}\times156\times20 = 52\times20 = 1040 \). For Y: \( B_Y = 12\times10 = 120 \), \( V_Y=\frac{1}{3}\times120\times h = 40h \). Set equal: \( 40h = 1040 \), so \( h = 26 \)? But wait, the options have D as 26. Wait, but let me check again. Wait, maybe the base of X: wait, the diagram: Pyramid X has base 13 and 12, height 20. Pyramid Y has base 12 and 10, height h. So yes, calculation: \( \frac{1}{3}\times13\times12\times20 = \frac{1}{3}\times12\times10\times h \). Cancel \( \frac{1}{3} \): \( 13\times12\times20 = 12\times10\times h \). Divide both sides by 12: \( 13\times20 = 10\times h \). So \( 260 = 10h \), so \( h = 26 \). Oh! I see, I made a mistake earlier in canceling. So \( 13\times12\times20 = 12\times10\times h \), divide both sides by 12: \( 13\times20 = 10h \), so \( h = \frac{260}{10}=26 \). So the height is 26 cm.

Answer:

D. 26 cm