Without using a calculator, explain why the solution to each equation must be a real number between 1 and 2.
21^x = 30

Explanation:

  21^1 = 21 < 30

  21^2 = 21² = 441 > 30

30 lies between 21^1 and 21^2.

21^x is a continuous function, so the value of x that makes 21^x = 30 must lie between 1 and 2, according to the intermediate value theorem.

_____

Using a calculator, we find x ≈ 1.117153.

option a, c,d are correct.

step-by-step explanation:

from the given figure, it is given that z is equidistant from the sides of the triangle rst, then from triangle tzb and triangle szb, we have

tz=sz(given)

bz=zb(common)

therefore, by rhs rule,δtzb ≅δszb

by cpctc, sz≅tz

also, from δctz and δasz,

tz=sz(given)

∠tcz=∠saz(90°)

by rhs rule, δctz ≅ δasz, therefore by cpctc, ∠ctz≅∠asz

also,from δasz and δzsb,

zs=sz(common)

∠zbs=∠saz=90°

by rhs rule, δasz ≅δzsb, therefore, by cpctc, ∠asz≅∠zsb

hence, option a, c,d are correct.

hard

step-by-step explanation: