Question

Find the height $h$ of a pyramid if base radius $r = 5 cm$ and total surface $A = 125 cm$ .

Answer

Problem has no solutions.

Explanation

STEP 1: find side $a$

To find side $a$ use formula:

r = \frac{a}{2}

After substituting $r = 5 cm$ we have:

5 cm = \frac{a}{2}

a = 5 cm \cdot 2

a = 10 cm

STEP 2: find base area $B$

To find base area $B$ use formula:

B = a^{2}

After substituting $a = 10 cm$ we have:

B = (10 cm)^{2}

B = 100 cm^{2}

STEP 3: find lateral surface $L$

To find lateral surface $L$ use formula:

A = B + L

After substituting $A = 125 cm$ and $B = 100 cm^{2}$ we have:

125 cm = 100 cm^{2} + L

L = 125 cm - 100 cm^{2}

L = 25 cm

STEP 4: find slant height $s$

To find slant height $s$ use formula:

L = 2 \cdot a \cdot s

After substituting $L = 25 cm$ and $a = 10 cm$ we have:

25 cm = 2 \cdot 10 cm \cdot s

25 cm = 20 cm \cdot s

s = \frac{25 cm}{20 cm}

s = \frac{5}{4}

STEP 5: find height $h$

To find height $h$ use Pythagorean Theorem:

h^{2} + \frac{a ^{2}}{4} = s^{2}

After substituting $a = 10 cm$ and $s = \frac{5}{4} cm^{0}$ we have:

h^{2} + \frac{( 10 cm ) ^{2}}{4} = \frac{5}{4}

h^{2} + \frac{100 cm ^{2}}{4} = \frac{5}{4}

h^{2} + 25 cm^{2} = \frac{5}{4}

h^{2} = \frac{25}{16} - 25 cm^{2}

h^{2} = - \frac{375}{16}

This equation has no solution $⟹$ The problem has no solution.

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