Question

name
grade 8 chapter 1 chapter practice test
date
14 the formula (k=\frac{5}{9}(f - 32)+273.15) converts temperatures from degrees fahrenheit f to kelvin k. an object in a laboratory is cooled to 25 kelvin. what is the temperature of the object in degrees fahrenheit?
temperature
15 the circles are identical. find the circumference of each circle. express your answer in terms of (pi) or rounded to the nearest tenth place. (c = pi d)

Explanation:

Step1: Solve for Fahrenheit in temperature - conversion formula

Given $K=\frac{5}{9}(F - 32)+273.15$, first isolate the term with $F$.
Subtract 273.15 from both sides: $K - 273.15=\frac{5}{9}(F - 32)$.
Then multiply both sides by $\frac{9}{5}$: $\frac{9}{5}(K - 273.15)=F - 32$.
Finally, add 32 to both sides: $F=\frac{9}{5}(K - 273.15)+32$.

Step2: Substitute $K = 25$ into the formula

$F=\frac{9}{5}(25 - 273.15)+32$.
First, calculate inside the parentheses: $25-273.15=-248.15$.
Then, multiply by $\frac{9}{5}$: $\frac{9}{5}\times(-248.15)=9\times(-49.63)=-446.67$.
Finally, add 32: $F=-446.67 + 32=-414.67$.

Step3: Solve for $x$ in circle - radius and diameter relationship

Since the circles are identical, the radius of the first circle times 2 is equal to the diameter of the second circle. So, $2\times6x=10x + 4$.
Expand the left - hand side: $12x=10x + 4$.
Subtract $10x$ from both sides: $12x-10x=4$, which gives $2x = 4$.
Divide both sides by 2: $x = 2$.

Step4: Calculate the circumference of the circle

The formula for the circumference of a circle is $C=\pi d$.
Using the diameter of the second circle $d = 10x+4$, substitute $x = 2$ into it, $d=10\times2 + 4=24$.
So, $C = 24\pi$. If we approximate $\pi\approx3.14$, then $C\approx24\times3.14 = 75.4$.

Answer:

1. $-414.67^{\circ}F$
2. $24\pi$ (or approximately $75.4$)