Substitution

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#13881

The formula for a summer night’s temperature (in degrees Celsius) is T = ^C/₈ + 3, where C is the number of chips per minute made by a cricket. How many chips per minute are made when the temperature is 13°C?

13 = ^C/₈ + 3            (subtract 3 from both sides)   

 10 = ^C/₈    (multiply both sides by 9

 80 = C

∴ 80 chirps per min

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#13882

The size of each angle in a regular polygon (in degrees) is $$a=180-\frac{360}{n}$$_. How many sides has the regular polygon with equal angles of 140°?

$$140=180-\frac{360}{n}$$ (-180 from both sides)

-180 -40 = – ³⁶⁰/_n

40 = ³⁶⁰/_n

40n = 360

n = ³⁶⁰/₄₀

n = 9

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#13883

Given the formula   b² = a²(1 – e²), evaluate e when a = 2.6 × 10¹⁵ and b = 7.3 × 10¹⁴, giving your answer in scientific notation to 5 significant figures. 

(7.3 × 10¹⁴)² = (2.6 × 10¹⁵)²(1 – e²)

##\frac{(7.3\times10^{14})^2}{(2.6\times10^{15})^2}=1-e^2##

##e^2+\frac{(7.3\times10^{14})^2}{(2.6\times10^{15})^2}=1##

##e^2=1-\frac{(7.3\times10^{14})^2}{(2.6\times10^{15})^2}##

##e = \sqrt {1 – \frac{{{{(7.3 \times {{10}^{14}})}^2}}}{{{{(2.6 \times {{10}^{15}})}^2}}}} ##

e = 9.5978 × 10^-1

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#18963

If $$I=\frac{E}{R+r}$$. find r if I = 12, E = 60 and R = 4.
$$12=\frac{60}{4+r}$$	substitute all numbers into their corresponding letters
$$12(4+r)=\frac{60}{\cancel{(4+r)}}^{\times\cancel{(4+r)}}$$	multiply both sides by 4 + r
48 + 12r = 60	expand
12r = 12	subtract 48 from both sides
r = 1	divide both sides by 12

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#18964

The sum of a geometric sequence is:  $$S=\frac{a(1-r^n)}{1-r}$$.  Find a when r = 0.8, n = 5 and S = 10.
$$10=\frac{a(1-0.8^5)}{1-0.8}$$	substitute all numbers into their corresponding letters
$$10=\frac{a(1-0.8^5)}{0.2}$$	simplify where possible
$$10\times0.2=\frac{a(1-0.8^5)}{\cancel{0.2}}^{\times\cancel{0.2}}$$	multiply both sides by 0.2
2 = a(1 – 0.85)	divide both sides by (1 – 0.85)
$$a=\frac{2}{1-0.8^5}$$	calculate
a = 2.97

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#18965

Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v kilometres per hour is given by:  P(v) = 0.015v3. How fast  must the wind blow in order to generate 120 watts of power in 1 hr?
120 = 0.015v3	substitute P(v) = 120
8000 = v3	solve the equation to find v (the speed)
v = 3√8000	take the cube root of both sides
v = 20

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