Ask a New Question
Menu

Use the image to answer the question.

An illustration shows a square pyramid with labeled parts. The left, front, and right faces of the pyramid and base are visible. The back face is not visible, but its position is indicated by a dashed line. The back base edge is not visible, but its position is indicated by a dashed line. The left base edge is labeled a. The back left lateral edge is labeled e. A dotted line extends from the interior of the vertex straight down to the center of the base and is labeled h. A right angle is indicated where the dotted height line meets the base. Another dotted line extends from the center of the base where the height line meets to the center of the right base edge and is labeled r. A dotted line extends from the right base edge straight up the center of the right face of the prism and meets the height line at the vertex. The dotted slant height line is labeled s.

For the square pyramid, the lateral edge length, e
, is 25 in., and the slant height, s
, is 24 in. Find the height of the pyramid, h
. Round the answer to the nearest whole number. Hint: Before finding the length of h
, you will first need to find the length of r
.

2 answers

To find the height \( h \) of the pyramid, we can use the relationship between the slant height \( s \), the height \( h \), and the radius \( r \) (which is half of the base edge length) of the pyramid.

  1. Understanding the Pyramid Geometry:

    • The slant height \( s \) is the distance from the vertex of the pyramid to the midpoint of any side of the base.
    • The height \( h \) is the vertical distance from the vertex down to the center of the base.
    • The radius \( r \) represents half of the base edge, thus is the horizontal distance from the center of the base to the midpoint of the base edge adjacent to the slant height.

  2. Using the Pythagorean Theorem: The relationship between these three components is given by the Pythagorean theorem: \[ s^2 = h^2 + r^2 \]

  3. Finding \( r \): Since \( r \) is half the length of the base edge but is not provided directly, we need to calculate \( r \) using the lateral edge \( e \) of the pyramid. It's important to note that if we label \( b \) as the length of the entire base side: \[ r = \frac{b}{2} \]

    From the right triangle formed with \( h \), \( r \), and \( e \): \[ e^2 = h^2 + r^2 \] Substituting the known values, we have \( e = 25 \) inches.

  4. Setting Up the Equations: We will substitute \( r = \frac{b}{2} \) into the previous equations later. For now, let's focus on the existing relationship with the known slant height \( s \) which is \( 24 \) inches: \[ s^2 = h^2 + r^2 \implies 24^2 = h^2 + r^2 \implies 576 = h^2 + r^2 \tag{1} \]

Also, from the lateral edge relation with height: \[ e^2 = h^2 + r^2 \implies 25^2 = h^2 + r^2 \implies 625 = h^2 + r^2 \tag{2} \]

  1. Subtracting the Two Equations: Now we proceed by subtracting equation (1) from equation (2): \[ 625 - 576 = (h^2 + r^2) - (h^2 + r^2) \implies 625 - 576 = 49 \]

This indicates that \( r^2 \) cancels out, leading us closer to solve for just \( h \).

  1. Finding \( h \): Another approach would be: \[ (25^2 - h^2) = (24^2 - h^2) \]

Assuming both satisfying relationships and equal radii contribute to distinct solutions.

Setting \( r \) to geometrical half values corresponding lends \( h \) solvability.

  1. Final Value Calculation: Let’s directly calculate \( h \): \[ 625 = h^2 + r^2 \ \implies h^2 = 625 - 576 = 49 \ \implies h = \sqrt{49} = 7 \]

Thus, the height of the pyramid \( h \) is approximately:

Final Answer:

7 inches (rounded to the nearest whole number).

incorrect