TASK 3 Direction: Solve for the indicated variable in each of the following: Example: If x varies directly as y and x = 12 when y = 2, what is the value of y when x = 24? Solution: Since x varies directly as y, then the equation of variation is in the form x = ky. Substitute the given values of x and y to solve for k in the equation. X = ky 12=k(2)—-divide both sides by 2 K= 6. Hence the equation of variation is x = 6y. Solving for y when x = 24. X=6y 24=6y —-divides both sides by 6 Y=4. Hence the value of y is 4 or y=4. 1. If x varies directly as y and x = 35 when y = 7, what is the value of y when x = 25? 2. If y varies inversely as x and y = 3 when x = 4, find y when x = 6 3. Z varies jointly as x and y. if z = 3 when x = 3 and y = 15,
The Correct Answer and Explanation is:
Problem 1: If xxx varies directly as yyy and x=35x = 35x=35 when y=7y = 7y=7, what is the value of yyy when x=25x = 25x=25?
Step 1: Since xxx varies directly as yyy, we can write the equation of variation as:x=kyx = kyx=ky
where kkk is the constant of proportionality.
Step 2: Use the given values x=35x = 35x=35 and y=7y = 7y=7 to find kkk.35=k×735 = k \times 735=k×7
Divide both sides by 7:k=357=5k = \frac{35}{7} = 5k=735=5
Step 3: Now that we know k=5k = 5k=5, we can write the equation of variation as:x=5yx = 5yx=5y
Step 4: To find yyy when x=25x = 25x=25, substitute x=25x = 25x=25 into the equation:25=5y25 = 5y25=5y
Divide both sides by 5:y=255=5y = \frac{25}{5} = 5y=525=5
Answer for Problem 1: The value of yyy is 5.
Problem 2: If yyy varies inversely as xxx and y=3y = 3y=3 when x=4x = 4x=4, find yyy when x=6x = 6x=6.
Step 1: Since yyy varies inversely as xxx, we can write the equation of variation as:y=kxy = \frac{k}{x}y=xk
where kkk is the constant of proportionality.
Step 2: Use the given values y=3y = 3y=3 and x=4x = 4x=4 to find kkk.3=k43 = \frac{k}{4}3=4k
Multiply both sides by 4:k=3×4=12k = 3 \times 4 = 12k=3×4=12
Step 3: Now that we know k=12k = 12k=12, we can write the equation of variation as:y=12xy = \frac{12}{x}y=x12
Step 4: To find yyy when x=6x = 6x=6, substitute x=6x = 6x=6 into the equation:y=126=2y = \frac{12}{6} = 2y=612=2
Answer for Problem 2: The value of yyy is 2.
Problem 3: zzz varies jointly as xxx and yyy. If z=3z = 3z=3 when x=3x = 3x=3 and y=15y = 15y=15, find the equation of variation and use it to find zzz for given values of xxx and yyy.
Step 1: Since zzz varies jointly as xxx and yyy, the equation of variation is:z=kxyz = kxyz=kxy
where kkk is the constant of proportionality.
Step 2: Use the given values z=3z = 3z=3, x=3x = 3x=3, and y=15y = 15y=15 to find kkk.3=k×3×153 = k \times 3 \times 153=k×3×15
Simplify the right-hand side:3=k×453 = k \times 453=k×45
Now, divide both sides by 45:k=345=115k = \frac{3}{45} = \frac{1}{15}k=453=151
Step 3: Now that we know k=115k = \frac{1}{15}k=151, the equation of variation becomes:z=115xyz = \frac{1}{15}xyz=151xy
Step 4: To find zzz for any values of xxx and yyy, simply substitute the values of xxx and yyy into the equation. For example, if x=4x = 4x=4 and y=10y = 10y=10:z=115×4×10=4015=83z = \frac{1}{15} \times 4 \times 10 = \frac{40}{15} = \frac{8}{3}z=151×4×10=1540=38
Answer for Problem 3: The equation of variation is z=115xyz = \frac{1}{15}xyz=151xy, and for x=4x = 4x=4 and y=10y = 10y=10, z=83z = \frac{8}{3}z=38.
