Question 1.
The Pythagoras Theorem states that the square of the hypotenuse is equal to the sum of the square of the two legs.
[tex]h^2=l_1^2+l_2^2[/tex]
From the question, the hypotenuse is h=10 yd and one of the legs is 2yd.
This implies that,
[tex]10^2=l_1^2+ {2}^{2} [/tex]
Evaluate the exponents
[tex]100=l_1^2+ 4[/tex]
Subbtract 4 from each side
[tex]100 - 4=l_1^2[/tex]
[tex]l_1^2 = 96[/tex]
[tex]l_1= \sqrt{96} [/tex]
[tex]l_1 = \sqrt{16 \times 6} [/tex]
[tex]l_1 = 4\sqrt{6} [/tex]
The correct answer is C
Question 2
We want to solve
[tex] {e}^{4x} = 11[/tex]
using inverse functions.
The inverse of an exponential function is a logarithmic function.
We take natural log of both sides to get;
[tex] ln({e}^{4x}) = \ln(11) [/tex]
Recall that
[tex] \ln( {a}^{n} ) = n \ln(a) [/tex]
This means
[tex]4x \ln({e}) = \ln(11) [/tex]
Logarithm of the base is 1
[tex]4x (1) = \ln(11) [/tex]
[tex]4x = \ln(11) [/tex]
Divide both sides by 4
[tex]x= \frac{ \ln(11)}{4} \\x=0.5995\\ x = 0.6[/tex]
